message stringlengths 2 57.2k | message_type stringclasses 2 values | message_id int64 0 1 | conversation_id int64 61 108k | cluster float64 22 22 | __index_level_0__ int64 122 217k |
|---|---|---|---|---|---|
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,029 | 22 | 80,058 |
Tags: brute force, math, number theory
Correct Solution:
```
def check(n):
d = 2
while d*d <= n:
if n % d == 0: return False
d += 1
return True
n = int(input())
flag = False
if check(n) and n > 10: print(1,'\n',n)
elif n >= 9:
Ans = [3]
n -= 3
for i in range(2,n):
if check(i) and check(n-i):
print(3)
print(3, i , n-i)
break
elif n == 3: print(1,'\n',3)
elif n == 5: print(1,'\n',5)
elif n == 7: print(1,'\n',7)
``` | output | 1 | 40,029 | 22 | 80,059 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,030 | 22 | 80,060 |
Tags: brute force, math, number theory
Correct Solution:
```
def isPrime(x) :
if x == 1: return False
if x == 2: return True
if x%2 == 0: return False
i = 3
while i*i <= x :
if x % i == 0 :
return False
i += 2
return True
n = int(input())
if isPrime(n) :
ans = [n]
else :
bigPrime = 0
for i in range(n-1, 0, -1) :
if isPrime(i) :
bigPrime = i
break
ans = [bigPrime]
n -= bigPrime
if isPrime(n) :
ans.append(n)
else :
for i in range(1, n) :
j = n - i
if isPrime(i) and isPrime(j) :
ans.append(i)
ans.append(j)
break
print(len(ans))
for i in ans :
print(i, end = ' ')
``` | output | 1 | 40,030 | 22 | 80,061 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,031 | 22 | 80,062 |
Tags: brute force, math, number theory
Correct Solution:
```
import math
def is_prime(a):
for i in range(2, int(math.sqrt(a) + 1)):
if a % i == 0:
return False
return True
def solve():
n, = map(int, input().split())
if is_prime(n):
print(1)
print(n)
return
number = n
while not is_prime(number):
number -= 1
n -= number
if is_prime(n):
print(2)
print(number, n)
return
for i in range(2, n):
if is_prime(i) and is_prime(n - i):
print(3)
print(number, i, n - i)
return
if __name__ == "__main__":
t = 1
# t = int(input())
for _ in range(t):
solve()
``` | output | 1 | 40,031 | 22 | 80,063 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,032 | 22 | 80,064 |
Tags: brute force, math, number theory
Correct Solution:
```
def isPrime(n) :
if (n==3 or n==2) :
return True
if (n % 2 == 0 or n % 3 == 0) :
return False
i = 5
while(i * i <= n) :
if (n % i == 0 or n % (i + 2) == 0) :
return False
i = i + 6
return True
n=int(input())
if isPrime(n):
print(1)
print(n)
exit()
print(3)
n-=3
for i in range(2,n):
if isPrime(i) and isPrime(n-i):
print(3,i,n-i)
exit()
``` | output | 1 | 40,032 | 22 | 80,065 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,033 | 22 | 80,066 |
Tags: brute force, math, number theory
Correct Solution:
```
l=list()
def prime(N):
count=0
i=1
while(i*i<=N):
if i*i==N:
count=count+1
elif N%i==0:
count=count+2
if count>2:
return 0
i=i+1
if count==2:
return 1
else:
return 0
n=int(input())
if n%2==0:
for j in range(2,n):
s=prime(j)
t=prime(n-j)
if s==1 and t==1:
l.append(j)
l.append(n-j)
break
else:
l.append(3)
if n-3==0:
l=[3]
elif n-3>2:
n=n-3
for j in range(2,n):
s=prime(j)
t=prime(n-j)
if s==1 and t==1:
l.append(j)
l.append(n-j)
break
else:
l=[5]
n=n-5
if n>2:
for j in range(2,n):
s=prime(j)
t=prime(n-j)
if s==1 and t==1:
l.append(j)
l.append(n-j)
break
print(len(l))
print(*l)
``` | output | 1 | 40,033 | 22 | 80,067 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,034 | 22 | 80,068 |
Tags: brute force, math, number theory
Correct Solution:
```
import math as m
n=int(input())
def prime(n):
if n==1:
return False
for j in range(2,int(m.sqrt(n))+1):
if n%j==0:
return False
return True
if prime(n):
print(1)
print(n)
else :
ans=[]
ans.append(3)
n-=3
for j in range(n,1,-1):
if prime(j) and prime(n-j):
ans.append(j)
ans.append(n-j)
break
print(3)
print(*ans)
``` | output | 1 | 40,034 | 22 | 80,069 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,035 | 22 | 80,070 |
Tags: brute force, math, number theory
Correct Solution:
```
######### ## ## ## #### ##### ## # ## # ##
# # # # # # # # # # # # # # # # # # #
# # # # ### # # # # # # # # # # # #
# ##### # # # # ### # # # # # # # # #####
# # # # # # # # # # # # # # # # # #
######### # # # # ##### # ##### # ## # ## # #
"""
PPPPPPP RRRRRRR OOOO VV VV EEEEEEEEEE
PPPPPPPP RRRRRRRR OOOOOO VV VV EE
PPPPPPPPP RRRRRRRRR OOOOOOOO VV VV EE
PPPPPPPP RRRRRRRR OOOOOOOO VV VV EEEEEE
PPPPPPP RRRRRRR OOOOOOOO VV VV EEEEEEE
PP RRRR OOOOOOOO VV VV EEEEEE
PP RR RR OOOOOOOO VV VV EE
PP RR RR OOOOOO VV VV EE
PP RR RR OOOO VVVV EEEEEEEEEE
"""
"""
Perfection is achieved not when there is nothing more to add, but rather when there is nothing more to take away.
"""
import sys
input = sys.stdin.readline
# from bisect import bisect_left as lower_bound;
# from bisect import bisect_right as upper_bound;
# from math import ceil, factorial;
def ceil(x):
if x != int(x):
x = int(x) + 1
return x
def factorial(x, m):
val = 1
while x>0:
val = (val * x) % m
x -= 1
return val
def fact(x):
val = 1
while x > 0:
val *= x
x -= 1
return val
# swap_array function
def swaparr(arr, a,b):
temp = arr[a];
arr[a] = arr[b];
arr[b] = temp;
## gcd function
def gcd(a,b):
if b == 0:
return a;
return gcd(b, a % b);
## lcm function
def lcm(a, b):
return (a * b) // gcd(a, b)
## nCr function efficient using Binomial Cofficient
def nCr(n, k):
if k > n:
return 0
if(k > n - k):
k = n - k
res = 1
for i in range(k):
res = res * (n - i)
res = res / (i + 1)
return int(res)
## upper bound function code -- such that e in a[:i] e < x;
def upper_bound(a, x, lo=0, hi = None):
if hi == None:
hi = len(a);
while lo < hi:
mid = (lo+hi)//2;
if a[mid] < x:
lo = mid+1;
else:
hi = mid;
return lo;
## prime factorization
def primefs(n):
## if n == 1 ## calculating primes
primes = {}
while(n%2 == 0 and n > 0):
primes[2] = primes.get(2, 0) + 1
n = n//2
for i in range(3, int(n**0.5)+2, 2):
while(n%i == 0 and n > 0):
primes[i] = primes.get(i, 0) + 1
n = n//i
if n > 2:
primes[n] = primes.get(n, 0) + 1
## prime factoriazation of n is stored in dictionary
## primes and can be accesed. O(sqrt n)
return primes
## MODULAR EXPONENTIATION FUNCTION
def power(x, y, p):
res = 1
x = x % p
if (x == 0) :
return 0
while (y > 0) :
if ((y & 1) == 1) :
res = (res * x) % p
y = y >> 1
x = (x * x) % p
return res
## DISJOINT SET UNINON FUNCTIONS
def swap(a,b):
temp = a
a = b
b = temp
return a,b;
# find function with path compression included (recursive)
# def find(x, link):
# if link[x] == x:
# return x
# link[x] = find(link[x], link);
# return link[x];
# find function with path compression (ITERATIVE)
def find(x, link):
p = x;
while( p != link[p]):
p = link[p];
while( x != p):
nex = link[x];
link[x] = p;
x = nex;
return p;
# the union function which makes union(x,y)
# of two nodes x and y
def union(x, y, link, size):
x = find(x, link)
y = find(y, link)
if size[x] < size[y]:
x,y = swap(x,y)
if x != y:
size[x] += size[y]
link[y] = x
## returns an array of boolean if primes or not USING SIEVE OF ERATOSTHANES
def sieve(n):
prime = [True for i in range(n+1)]
prime[0], prime[1] = False, False
p = 2
while (p * p <= n):
if (prime[p] == True):
for i in range(p * p, n+1, p):
prime[i] = False
p += 1
return prime
# Euler's Toitent Function phi
def phi(n) :
result = n
p = 2
while(p * p<= n) :
if (n % p == 0) :
while (n % p == 0) :
n = n // p
result = result * (1.0 - (1.0 / (float) (p)))
p = p + 1
if (n > 1) :
result = result * (1.0 - (1.0 / (float)(n)))
return (int)(result)
def is_prime(n):
for i in range(2, int(n ** (1 / 2)) + 1):
if not n % i:
return False
return True
#### PRIME FACTORIZATION IN O(log n) using Sieve ####
MAXN = int(1e5 + 5)
def spf_sieve():
spf[1] = 1;
for i in range(2, MAXN):
spf[i] = i;
for i in range(4, MAXN, 2):
spf[i] = 2;
for i in range(3, ceil(MAXN ** 0.5), 2):
if spf[i] == i:
for j in range(i*i, MAXN, i):
if spf[j] == j:
spf[j] = i;
## function for storing smallest prime factors (spf) in the array
################## un-comment below 2 lines when using factorization #################
spf = [0 for i in range(MAXN)]
# spf_sieve();
def factoriazation(x):
res = []
for i in range(2, int(x ** 0.5) + 1):
while x % i == 0:
res.append(i)
x //= i
if x != 1:
res.append(x)
return res
## this function is useful for multiple queries only, o/w use
## primefs function above. complexity O(log n)
def factors(n):
res = []
for i in range(1, int(n ** 0.5) + 1):
if n % i == 0:
res.append(i)
res.append(n // i)
return list(set(res))
## taking integer array input
def int_array():
return list(map(int, input().strip().split()));
def float_array():
return list(map(float, input().strip().split()));
## taking string array input
def str_array():
return input().strip().split();
#defining a couple constants
MOD = int(1e9)+7;
CMOD = 998244353;
INF = float('inf'); NINF = -float('inf');
################### ---------------- TEMPLATE ENDS HERE ---------------- ###################
from itertools import permutations
import math
from bisect import bisect_left
def solve():
n = int(input())
# primes = sieve(n)
if is_prime(n):
print(1)
print(n)
return
ans = []
i = n
c = 3
for i in range(n, 1, -1):
if is_prime(i):
ans.append(i)
n -= i
break
for i in range(2, n + 1):
if is_prime(i) and n - i == 0:
ans.append(i)
break
elif is_prime(i) and is_prime(n - i):
ans.append(i)
ans.append(n - i)
break
print(len(ans))
print(*ans)
if __name__ == '__main__':
for _ in range(1):
solve()
# fin_time = datetime.now()
# print("Execution time (for loop): ", (fin_time-init_time))
``` | output | 1 | 40,035 | 22 | 80,071 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,036 | 22 | 80,072 |
Tags: brute force, math, number theory
Correct Solution:
```
import math
n = int(input())
p = [0 for _ in range(1001)]
x = 2
p[0] = 1
p[1] = 1
while x * x <= 1001:
for i in range(2 * x, 1001, x):
p[i] = 1
x += 1
pr = 0
def isprime(num):
for i in range(2, int(math.sqrt(n)) + 1):
if num % i == 0:
return False
return True
e = 0
if isprime(n):
print(1)
print(n)
exit(0)
for k in range(n, 1, -1):
if isprime(k):
if (n - k) % 2 == 0:
pr = n - k
e = 1
break
if pr == 2:
print(2)
print(2, n - 2)
exit(0)
for i in range(2, 1001):
if p[pr - i] == 0 and p[i] == 0:
print(3)
print(n - pr, i, pr - i)
exit(0)
``` | output | 1 | 40,036 | 22 | 80,073 |
Provide tags and a correct Python 2 solution for this coding contest problem.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself. | instruction | 0 | 40,037 | 22 | 80,074 |
Tags: brute force, math, number theory
Correct Solution:
```
from sys import stdin, stdout
from collections import Counter, defaultdict
from itertools import permutations, combinations
raw_input = stdin.readline
pr = stdout.write
mod=10**9+7
def ni():
return int(raw_input())
def li():
return map(int,raw_input().split())
def pn(n):
stdout.write(str(n)+'\n')
def pa(arr):
pr(' '.join(map(str,arr))+'\n')
# fast read function for total integer input
def inp():
# this function returns whole input of
# space/line seperated integers
# Use Ctrl+D to flush stdin.
return map(int,stdin.read().split())
range = xrange # not for python 3.0+
# main code
def isPrime(x):
i=2
while i*i<=x:
if x%i==0:
return False
i+=1
return True
n=ni()
if isPrime(n):
print 1
print n
exit()
for i in range(2,10**6):
temp=n-3-i
if temp>1 and isPrime(i) and isPrime(temp):
print 3
print 3,i,temp
exit()
``` | output | 1 | 40,037 | 22 | 80,075 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
from sys import stdin
input=lambda : stdin.readline().strip()
lin=lambda :list(map(int,input().split()))
iin=lambda :int(input())
main=lambda :map(int,input().split())
from math import ceil,sqrt,factorial,log
from collections import deque
from bisect import bisect_left
def isprime(x):
if x==3 or x==5:
return 1
for i in range(2,ceil(sqrt(x)+1)):
if x%i==0:
return 0
return 1
def solve():
n=iin()
l=[]
z=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973,10007,10009,10037,10039,10061,10067,10069,10079,10091,10093,10099,10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193,10211,10223,10243,10247,10253,10259,10267,10271,10273,10289,10301,10303,10313,10321,10331,10333,10337,10343,10357,10369,10391,10399,10427,10429,10433,10453,10457,10459,10463,10477,10487,10499,10501,10513,10529,10531,10559,10567,10589,10597,10601,10607,10613,10627,10631,10639,10651,10657,10663,10667,10687,10691,10709,10711,10723,10729,10733,10739,10753,10771,10781,10789,10799,10831,10837,10847,10853,10859,10861,10867,10883,10889,10891,10903,10909,10937,10939,10949,10957,10973,10979,10987,10993,11003,11027,11047,11057,11059,11069,11071,11083,11087,11093,11113,11117,11119,11131,11149,11159,11161,11171,11173,11177,11197,11213,11239,11243,11251,11257,11261,11273,11279,11287,11299,11311,11317,11321,11329,11351,11353,11369,11383,11393,11399,11411,11423,11437,11443,11447,11467,11471,11483,11489,11491,11497,11503,11519,11527,11549,11551,11579,11587,11593,11597,11617,11621,11633,11657,11677,11681,11689,11699,11701,11717,11719,11731,11743,11777,11779,11783,11789,11801,11807,11813,11821,11827,11831,11833,11839,11863,11867,11887,11897,11903,11909,11923,11927,11933,11939,11941,11953,11959,11969,11971,11981,11987,12007,12011,12037,12041,12043,12049,12071,12073,12097,12101,12107,12109,12113,12119,12143,12149,12157,12161,12163,12197,12203,12211,12227,12239,12241,12251,12253,12263,12269,12277,12281,12289,12301,12323,12329,12343,12347,12373,12377,12379,12391,12401,12409,12413,12421,12433,12437,12451,12457,12473,12479,12487,12491,12497,12503,12511,12517,12527,12539,12541,12547,12553,12569,12577,12583,12589,12601,12611,12613,12619,12637,12641,12647,12653,12659,12671,12689,12697,12703,12713,12721,12739,12743,12757,12763,12781,12791,12799,12809,12821,12823,12829,12841,12853,12889,12893,12899,12907,12911,12917,12919,12923,12941,12953,12959,12967,12973,12979,12983,13001,13003,13007,13009,13033,13037,13043,13049,13063,13093,13099,13103,13109,13121,13127,13147,13151,13159,13163,13171,13177,13183,13187,13217,13219,13229,13241,13249,13259,13267,13291,13297,13309,13313,13327,13331,13337,13339,13367,13381,13397,13399,13411,13417,13421,13441,13451,13457,13463,13469,13477,13487,13499,13513,13523,13537,13553,13567,13577,13591,13597,13613,13619,13627,13633,13649,13669,13679,13681,13687,13691,13693,13697,13709,13711,13721,13723,13729,13751,13757,13759,13763,13781,13789,13799,13807,13829,13831,13841,13859,13873,13877,13879,13883,13901,13903,13907,13913,13921,13931,13933,13963,13967,13997,13999,14009,14011,14029,14033,14051,14057,14071,14081,14083,14087,14107,14143,14149,14153,14159,14173,14177,14197,14207,14221,14243,14249,14251,14281,14293,14303,14321,14323,14327,14341,14347,14369,14387,14389,14401,14407,14411,14419,14423,14431,14437,14447,14449,14461,14479,14489,14503,14519,14533,14537,14543,14549,14551,14557,14561,14563,14591,14593,14621,14627,14629,14633,14639,14653,14657,14669,14683,14699,14713,14717,14723,14731,14737,14741,14747,14753,14759,14767,14771,14779,14783,14797,14813,14821,14827,14831,14843,14851,14867,14869,14879,14887,14891,14897,14923,14929,14939,14947,14951,14957,14969,14983]
s=set(z)
x=0
while n-2*x>0:
if isprime(n-2*x):
t=2*x
if t!=0:
for i in z:
if t-i==0:
l=[n-2*x,i]
break
elif t-i in s:
l=[n-2*x,i,t-i]
break
else:
l=[n]
break
x+=1
print(len(l))
print(*l)
# print(isprime(61))
qwe=1
# qwe=int(input())
for _ in range(qwe):
solve()
``` | instruction | 0 | 40,038 | 22 | 80,076 |
Yes | output | 1 | 40,038 | 22 | 80,077 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
gap = 300
m = 1 << 15
isp = [False]*2 + [True]*(m - 1)
ps = []
for i in range(2, m + 1):
if isp[i]:
ps.append(i)
for j in range(i * i, m + 1, i):
isp[j] = False
n = int(input())
assert n % 2
if n >= m:
low = n - gap
ispg = [True]*(gap + 1)
for p in ps:
for i in range((low - 1) // p + 1, n // p + 1):
ispg[i * p - low] = False
def is_prime(k):
return isp[k] if k <= m else ispg[k - low]
p = n
while not is_prime(p): p -= 2
s = [p]
if p != n:
if n - p == 2:
s += [n - p]
else:
q = 2
while not (is_prime(q) and is_prime(n - p - q)): q += 1
s += [q, n - p - q]
print(len(s))
print(*s)
``` | instruction | 0 | 40,039 | 22 | 80,078 |
Yes | output | 1 | 40,039 | 22 | 80,079 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
from math import sqrt
from array import array
from sys import stdin
input = stdin.readline
n = int(input())
ans = []
if n == 3:
ans.append(3)
print('{}'.format(len(ans)) + '\n' + ''.join(str(x) + ' ' for x in ans))
exit(0)
N = n
magic_value = 300
primes = array('L')
primes.append(2)
lim = int(sqrt(N)) + 1
for number in range(3, lim, 2):
isPrime = True
for prime in primes:
if number % prime == 0:
isPrime = False
break
if isPrime:
primes.append(number)
def checkPrime(x):
lim = int(sqrt(x) + 1)
isPrime = True
for prime in primes:
if x % prime == 0:
isPrime = False
break
if prime > lim:
break
return isPrime
for sub in range(0, magic_value, 2):
if checkPrime(n - sub):
ans.append(n - sub)
n = sub
break
if n == 0:
print('{}'.format(len(ans)) + '\n' + ''.join(str(x) + ' ' for x in ans))
exit(0)
for prime in primes:
if n == prime:
ans.append(n)
break
if n - prime in primes:
ans.append(prime)
ans.append(n - prime)
break
if prime > n:
break
print('{}'.format(len(ans)) + '\n' + ''.join(str(x) + ' ' for x in ans))
``` | instruction | 0 | 40,040 | 22 | 80,080 |
Yes | output | 1 | 40,040 | 22 | 80,081 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
def main():
n = int(input())
if n == 3:
print(1)
print(3)
elif n == 4:
print(2)
print(2, 2)
elif n == 5:
print(2)
print(2, 3)
elif n == 6:
print(2)
print(3, 3)
elif n == 7:
print(2)
print(2, 5)
elif n == 8:
print(2)
print(3, 5)
elif n == 9:
print(2)
print(2, 7)
elif n == 10:
print(2)
print(3, 7)
if n <= 10:
return
for i in range(n, -1, -1):
j = 2
simple = True
while (j * j <= i):
if (i % j == 0):
simple = False
j += 1
if simple:
n -= i
break
ans = [i]
prime = [True] * (n+2)
prime[0] = False
prime[1] = False
#print(prime)
for i in range(2, n+1):
if prime[i]:
if i * i <= n:
for j in range(i * i, n+1, i):
#print(j)
prime[j] = False
#print(i, prime)
i = k = 0
for i in range(2, n+1):
k = n - i
if prime[i] and prime[k]:
break
if i != 0:
ans.append(i)
if k != 0:
ans.append(k)
print(len(ans))
for x in ans:
print(x, end=" ")
main()
``` | instruction | 0 | 40,041 | 22 | 80,082 |
Yes | output | 1 | 40,041 | 22 | 80,083 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
from math import *
prime = [ 0,2 , 3 , 5 , 7 , 11 , 13, 17, 19, 23, 29,
31 , 37 , 41 , 43 , 47 , 53 , 59, 61, 67, 71,
73 , 79 , 83 , 89 , 97 , 101 , 103, 107, 109 , 113,
127 , 131 , 137 , 139 , 149 , 151 , 157 , 163 , 167, 173,
179 , 181 , 191 , 193 , 197 , 199 , 211 , 223 , 227 , 229,
233 , 239 , 241 , 251 , 257 , 263 , 269 , 271 , 277 , 281,
283 , 293 , 307 , 311 , 313 , 317 , 331 , 337 , 347 , 349,
353 , 359 , 367 , 373 , 379 , 383 , 389 , 397 , 401 , 409,
419 , 421 , 431 , 433 , 439 , 443 , 449 , 457 , 461 , 463,
467 , 479 , 487 , 491 , 499 , 503 , 509 , 521 , 523 , 541,
547 , 557 , 563 , 569 , 571 , 577 , 587 , 593 , 599 , 601,
607 , 613 , 617 , 619 , 631 , 641 , 643 , 647 , 653 , 659,
661 , 673 , 677 , 683 , 691 , 701 , 709 , 719 , 727 , 733,
739 , 743 , 751 , 757 , 761 , 769 , 773 , 787 , 797 , 809,
811 , 821 , 823 , 827 , 829 , 839 , 853 , 857 , 859 , 863,
877 , 881 , 883 , 887 , 907 , 911 , 919 , 929 , 937 , 941,
947 , 953 , 967 , 971, 977, 983 , 991, 997, 1009, 1013]
def divider(n):
if n==1:
return 0
i = 2
j = 0
while i**2 <= n and j != 1:
if n%i == 0:
j = 1
i += 1
if j == 1:
return 0
else:
return 1
n = int(input())
if n==3:
print(3)
quit()
for i in range(len(prime)):
for j in range(len(prime)):
if divider(n-prime[i]-prime[j])==1:
if prime[j]==0 and prime[i]==0:
print(n)
quit()
if prime[i]==0:
print(prime[j],n-prime[j])
quit()
if prime[j]==0:
print(prime[i],n-prime[i])
quit()
print(prime[i],prime[i],prime[j])
quit()
``` | instruction | 0 | 40,042 | 22 | 80,084 |
No | output | 1 | 40,042 | 22 | 80,085 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
#rOkY
#FuCk
############################### kOpAl ##################################
t=int(input())
print(1,1,t-2)
``` | instruction | 0 | 40,043 | 22 | 80,086 |
No | output | 1 | 40,043 | 22 | 80,087 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
from sys import stdin
input=lambda : stdin.readline().strip()
lin=lambda :list(map(int,input().split()))
iin=lambda :int(input())
main=lambda :map(int,input().split())
from math import ceil,sqrt,factorial,log
from collections import deque
from bisect import bisect_left
def isprime(x):
if x==3 or x==5:
return 1
for i in range(2,ceil(sqrt(x)+1)):
if x%i==0:
return 0
return 1
def solve():
n=iin()
if isprime(n):
print(1)
print(n)
elif isprime(n-2):
print(2)
print(n-2,2)
else:
print(3)
print(n-4,2,2)
qwe=1
# qwe=int(input())
for _ in range(qwe):
solve()
``` | instruction | 0 | 40,044 | 22 | 80,088 |
No | output | 1 | 40,044 | 22 | 80,089 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Dima loves representing an odd number as the sum of multiple primes, and Lisa loves it when there are at most three primes. Help them to represent the given number as the sum of at most than three primes.
More formally, you are given an odd numer n. Find a set of numbers pi (1 ≤ i ≤ k), such that
1. 1 ≤ k ≤ 3
2. pi is a prime
3. <image>
The numbers pi do not necessarily have to be distinct. It is guaranteed that at least one possible solution exists.
Input
The single line contains an odd number n (3 ≤ n < 109).
Output
In the first line print k (1 ≤ k ≤ 3), showing how many numbers are in the representation you found.
In the second line print numbers pi in any order. If there are multiple possible solutions, you can print any of them.
Examples
Input
27
Output
3
5 11 11
Note
A prime is an integer strictly larger than one that is divisible only by one and by itself.
Submitted Solution:
```
def prost(n):
k=0
for i in range(2,n):
if n%i==0:
k+=1
if k==0:
return(0)
else:
return(1)
n=int(input())
np=n
bo=prost(n)
kol=0
sd=''
s=''
if bo==0:
print(1,n,sep='\n')
else:
for i in range(n-1,1,-1):
if np==0:
break
else:
bo=prost(i)
if bo==0:
np=n-i
kol+=1
sd+=str(i)+' '
while np>0:
if np>2:
for i in range(2,np):
bo=prost(i)
if bo==0:
if np-i>=0:
np=np-i
kol+=1
sd+=str(i)+' '
if prost(np)==0:
kol+=1
print(kol,sd+str(np),sep='\n')
np=0
break
else:
pass
if np!=0:
np=0
if np!=0:
np=n
n=n-1
sd=s
kol=0
``` | instruction | 0 | 40,045 | 22 | 80,090 |
No | output | 1 | 40,045 | 22 | 80,091 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given an image, that can be represented with a 2-d n by m grid of pixels. Each pixel of the image is either on or off, denoted by the characters "0" or "1", respectively. You would like to compress this image. You want to choose an integer k > 1 and split the image into k by k blocks. If n and m are not divisible by k, the image is padded with only zeros on the right and bottom so that they are divisible by k. Each pixel in each individual block must have the same value. The given image may not be compressible in its current state. Find the minimum number of pixels you need to toggle (after padding) in order for the image to be compressible for some k. More specifically, the steps are to first choose k, then the image is padded with zeros, then, we can toggle the pixels so it is compressible for this k. The image must be compressible in that state.
Input
The first line of input will contain two integers n, m (2 ≤ n, m ≤ 2 500), the dimensions of the image.
The next n lines of input will contain a binary string with exactly m characters, representing the image.
Output
Print a single integer, the minimum number of pixels needed to toggle to make the image compressible.
Example
Input
3 5
00100
10110
11001
Output
5
Note
We first choose k = 2.
The image is padded as follows:
001000
101100
110010
000000
We can toggle the image to look as follows:
001100
001100
000000
000000
We can see that this image is compressible for k = 2. | instruction | 0 | 40,113 | 22 | 80,226 |
Tags: brute force
Correct Solution:
```
def debug(*a):
return
print('[DEBUG]', *a)
is_prime = [False, False] + [True] * (2600-2)
for i in range(2, 2600):
if is_prime[i]:
for j in range(i+i, 2600, i):
is_prime[j] = False
n, m = map(int, input().split())
a = [None] * n
for i in range(n):
a[i] = input()
b = [None] * (n+1)
b[n] = [0] * (m+1)
for i in range(n-1, -1, -1):
b[i] = [0] * (m+1)
for j in range(m-1, -1, -1):
b[i][j] = b[i+1][j] + b[i][j+1] - b[i+1][j+1] + int(a[i][j])
# debug(b[i])
ans = None
for k in range(2, max(n, m)+1):
# if is_prime[k]:
# debug('k=', k)
kk = k*k
aa = 0
for i in range(0, n, k):
for j in range(0, m, k):
bb = b[i][j]
if i + k < n:
bb -= b[i+k][j]
if j + k < m:
bb -= b[i][j+k]
if i + k < n and j + k < m:
bb += b[i+k][j+k]
# debug(i, j, bb)
aa += min(bb, kk-bb)
# debug('aa=', aa)
if ans == None or aa < ans:
ans = aa
print(ans)
``` | output | 1 | 40,113 | 22 | 80,227 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given an image, that can be represented with a 2-d n by m grid of pixels. Each pixel of the image is either on or off, denoted by the characters "0" or "1", respectively. You would like to compress this image. You want to choose an integer k > 1 and split the image into k by k blocks. If n and m are not divisible by k, the image is padded with only zeros on the right and bottom so that they are divisible by k. Each pixel in each individual block must have the same value. The given image may not be compressible in its current state. Find the minimum number of pixels you need to toggle (after padding) in order for the image to be compressible for some k. More specifically, the steps are to first choose k, then the image is padded with zeros, then, we can toggle the pixels so it is compressible for this k. The image must be compressible in that state.
Input
The first line of input will contain two integers n, m (2 ≤ n, m ≤ 2 500), the dimensions of the image.
The next n lines of input will contain a binary string with exactly m characters, representing the image.
Output
Print a single integer, the minimum number of pixels needed to toggle to make the image compressible.
Example
Input
3 5
00100
10110
11001
Output
5
Note
We first choose k = 2.
The image is padded as follows:
001000
101100
110010
000000
We can toggle the image to look as follows:
001100
001100
000000
000000
We can see that this image is compressible for k = 2. | instruction | 0 | 40,114 | 22 | 80,228 |
Tags: brute force
Correct Solution:
```
def debug(*a):
return
print('[DEBUG]', *a)
is_prime = [False, False] + [True] * (2600-2)
for i in range(2, 2600):
if is_prime[i]:
for j in range(i+i, 2600, i):
is_prime[j] = False
n, m = map(int, input().split())
a = [None] * n
for i in range(n):
a[i] = input()
b = [None] * (n+1)
b[n] = [0] * (m+1)
for i in range(n-1, -1, -1):
b[i] = [0] * (m+1)
for j in range(m-1, -1, -1):
b[i][j] = b[i+1][j] + b[i][j+1] - b[i+1][j+1] + int(a[i][j])
# debug(b[i])
ans = None
for k in range(2, max(n, m)+1):
if is_prime[k]:
# debug('k=', k)
kk = k*k
aa = 0
for i in range(0, n, k):
for j in range(0, m, k):
bb = b[i][j]
if i + k < n:
bb -= b[i+k][j]
if j + k < m:
bb -= b[i][j+k]
if i + k < n and j + k < m:
bb += b[i+k][j+k]
# debug(i, j, bb)
aa += min(bb, kk-bb)
# debug('aa=', aa)
if ans == None or aa < ans:
ans = aa
print(ans)
``` | output | 1 | 40,114 | 22 | 80,229 |
Provide a correct Python 3 solution for this coding contest problem.
Let M be a positive integer.
You are given 2 N integers a_1, a_2, \ldots, a_{2 N}, where 0 \leq a_i < M for each i.
Consider dividing the 2 N integers into N pairs. Here, each integer must belong to exactly one pair.
We define the ugliness of a pair (x, y) as (x + y) \mod M. Let Z be the largest ugliness of the N pairs. Find the minimum possible value of Z.
Constraints
* All values in input are integers.
* 1 \leq N \leq 10^5
* 1 \leq M \leq 10^9
* 0 \leq a_i < M
Input
Input is given from Standard Input in the following format:
N M
a_1 a_2 \cdots a_{2N}
Output
Print the minimum possible value of Z, where Z is the largest ugliness of the N pairs.
Examples
Input
3 10
0 2 3 4 5 9
Output
5
Input
2 10
1 9 1 9
Output
0 | instruction | 0 | 40,240 | 22 | 80,480 |
"Correct Solution:
```
def modulo_pairing(N: int, M: int, A: list)->int:
A.sort()
def check(n: int)->bool:
'''2*n の位置を区切りとして、右側が全て赤のペアになることを確認する。
'''
for i in range(2*n, N+n):
if A[i] + A[2*N+2*n-i-1] < M:
return False
return True
l, r = -1, N+1
while r - l > 1:
m = (l + r) // 2
if check(m):
r = m
else:
l = m
blue = [A[i]+A[2*r-i-1] for i in range(r)]
red = [A[i]+A[2*N+2*r-i-1]-M for i in range(2*r, N+r)]
return max(blue + red)
if __name__ == "__main__":
N, M = map(int, input().split())
A = [int(s) for s in input().split()]
ans = modulo_pairing(N, M, A)
print(ans)
``` | output | 1 | 40,240 | 22 | 80,481 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,294 | 22 | 80,588 |
"Correct Solution:
```
m=10**9+7;f=max;r=range
for _ in r(int(input())):
x,y=sorted(map(int,input().split()));s=-1;i=j=1
while j<=x and i+j<=y:i,j,s=j,i+j,s+1
if x<2 or y<3:print(1,x*y%m);continue
a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1);i=j=1
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
a+=(y>=k)*f(0,(x-l)//k+1)+(x>=k)*f(0,(y-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | output | 1 | 40,294 | 22 | 80,589 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,295 | 22 | 80,590 |
"Correct Solution:
```
from math import gcd
import random,time
def gcdcount(n,m):
x,y=min(n,m),max(n,m)
if x==0:
return 0
else:
return 1+gcdcount(x,y%x)
fibo=[0,1,2]
for i in range(100):
fibo.append(fibo[-1]+fibo[-2])
gcdfibo=[[],[(1,2)],[(2,3)],[(3,5),(4,7)]]
for i in range(4,101):
temp=[]
for a,b in gcdfibo[-1]:
temp.append((b,a+b))
b=gcdfibo[-1][1][1]
a=fibo[i]
temp.append((b,a+b))
temp.sort()
gcdfibo.append(temp)
def solve(x,y):
if x>y:
x,y=y,x
mod=10**9+7
if x==1:
return (1,y%mod)
if (x,y)==(2,2):
return (1,3)
id=1
t=-1
while True:
if fibo[id+1]<=x and fibo[id+2]<=y:
id+=1
else:
if fibo[id+1]>x:
if fibo[id+2]>y:
t=0
else:
t=1
else:
t=2
break
#print(t)
if t==0:
res=0
for a,b in gcdfibo[id]:
if a<=x and b<=y:
res+=1
return (id,res)
elif t==1:
res=0
for a,b in gcdfibo[id]:
if a<=x:
res+=(y-b)//a+1
res%=mod
return (id,res)
else:
res=0
for a,b in gcdfibo[id+1]:
u,v=b-a,a
if u<=x and v<=y:
res+=1
res%=mod
if u<=y and v<=x:
res+=1
res%=mod
return (id,res)
mod=10**9+7
Q=int(input())
start=time.time()
for _ in range(Q):
x,y=map(int,input().split())
id,res=solve(x,y)
if (x,y)==(2,2):
res+=1
print(id,res)
#print(time.time()-start)
``` | output | 1 | 40,295 | 22 | 80,591 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,296 | 22 | 80,592 |
"Correct Solution:
```
m=10**9+7;f=lambda a:max(0,(a-l)//k+1);r=range
for _ in r(int(input())):
x,y=sorted(map(int,input().split()));s=-1;i=j=k=l=1
while l<=x and k+l<=y:k,l,s=l,k+l,s+1;a=f(y)+f(x)
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
a,i,j=a+(y>=k)*f(x)+(x>=k)*f(y),j,i+j
if x<2 or y<3:print(1,x*y%m)
else:print(s+1,a%m)
``` | output | 1 | 40,296 | 22 | 80,593 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,297 | 22 | 80,594 |
"Correct Solution:
```
m=10**9+7
f=lambda x:max(x,0)
r=range
for _ in r(int(input())):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:i,j,s=j,i+j,s+1
if x==1 or y<3:print(1,x*y%m)
else:
a=f((y-j)//i+1)+f((x-j)//i+1)
i=j=1
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
if x>=k:a+=f((y-l)//k+1)
if y>=k:a+=f((x-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | output | 1 | 40,297 | 22 | 80,595 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,298 | 22 | 80,596 |
"Correct Solution:
```
m=10**9+7
f=max
r=range
for _ in r(int(input())):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:i,j,s=j,i+j,s+1
if x<2 or y<3:print(1,x*y%m);continue
else:
a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
i=j=1
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
if x>=k:a+=f(0,(y-l)//k+1)
if y>=k:a+=f(0,(x-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | output | 1 | 40,298 | 22 | 80,597 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,299 | 22 | 80,598 |
"Correct Solution:
```
m=10**9+7
f=max
r=range
for _ in r(int(input())):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:i,j,s=j,i+j,s+1
if x==1 or y<3:print(1,x*y%m)
else:
a=f(0,(y-j)//i+1)+f(0,(x-j)//i+1)
i=j=1
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
if x>=k:a+=f(0,(y-l)//k+1)
if y>=k:a+=f(0,(x-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | output | 1 | 40,299 | 22 | 80,599 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,300 | 22 | 80,600 |
"Correct Solution:
```
q=int(input())
m=10**9+7
for _ in range(q):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:
i,j=j,i+j
s+=1
if x==1 or y<3:print(1,x*y%m)
else:
a=max(0,(y-j)//i+1)+max(0,(x-j)//i+1)
i=j=1
for c in range(s):
k,l=j,i+j*2
for _ in range(s-c):k,l=l,k+l
if x>=k:a+=max(0,(y-l)//k+1)
if y>=k:a+=max(0,(x-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | output | 1 | 40,300 | 22 | 80,601 |
Provide a correct Python 3 solution for this coding contest problem.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2 | instruction | 0 | 40,301 | 22 | 80,602 |
"Correct Solution:
```
from math import gcd
import random,time
def gcdcount(n,m):
x,y=min(n,m),max(n,m)
if x==0:
return 0
else:
return 1+gcdcount(x,y%x)
fibo=[0,1,2]
for i in range(100):
fibo.append(fibo[-1]+fibo[-2])
gcdfibo=[[],[(1,2),(1,3)]]
for i in range(2,101):
temp=[]
for a,b in gcdfibo[-1]:
temp.append((b,a+b))
a,b=fibo[i],fibo[i+2]
temp.append((a,b))
temp.sort()
gcdfibo.append(temp)
def solve(x,y):
if x>y:
x,y=y,x
mod=10**9+7
if x==1:
return (1,y%mod)
if (x,y)==(2,2):
return (1,3)
id=1
t=-1
while True:
if fibo[id+1]<=x and fibo[id+2]<=y:
id+=1
else:
if fibo[id+1]>x:
if fibo[id+2]>y:
t=0
else:
t=1
else:
t=2
break
#print(t)
if t==0:
res=0
for a,b in gcdfibo[id]:
if a<=x and b<=y:
res+=1
return (id,res)
elif t==1:
res=0
for a,b in gcdfibo[id]:
if a<=x and (a,b)!=(fibo[id],fibo[id+2]) and b<=y:
res+=(y-b)//a+1
res%=mod
return (id,res)
else:
res=0
for a,b in gcdfibo[id+1]:
u,v=b-a,a
if u<=x and v<=y:
res+=1
res%=mod
if u<=y and v<=x:
res+=1
res%=mod
return (id,res)
mod=10**9+7
Q=int(input())
start=time.time()
for _ in range(Q):
x,y=map(int,input().split())
id,res=solve(x,y)
if (x,y)==(2,2):
res+=1
print(id,res)
#print(time.time()-start)
``` | output | 1 | 40,301 | 22 | 80,603 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
q = int(input())
mod = 10**9+7
for _ in range(q):
x, y = sorted(map(int, input().split()))
ma = 0
i, j = 1, 1
while j <= x and i+j <= y:
i, j = j, i+j
ma += 1
if x == 1 or y <= 2:
print(1, x*y % mod)
else:
ans = 0
if x >= i:
ans += max(0, (y-j) // i+1)
if y >= i:
ans += max(0, (x-j) // i+1)
i, j = 1, 1
for c in range(ma-1):
k, l = j, i+j*2
for _ in range(ma-c-1):
k, l = l, k+l
if x >= k:
ans += max(0, (y-l) // k+1)
if y >= k:
ans += max(0, (x-l) // k+1)
i, j = j, i+j
print(ma, ans % mod)
``` | instruction | 0 | 40,302 | 22 | 80,604 |
Yes | output | 1 | 40,302 | 22 | 80,605 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
m=10**9+7
f=max
r=range
p=print
for _ in r(int(input())):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:
i,j=j,i+j
s+=1
if x==1 or y<3:p(1,x*y%m)
else:
a=max(0,(y-j)//i+1)+f(0,(x-j)//i+1)
i=j=1
for c in r(s):
k,l=j,i+j*2
for _ in r(s-c):k,l=l,k+l
if x>=k:a+=f(0,(y-l)//k+1)
if y>=k:a+=f(0,(x-l)//k+1)
i,j=j,i+j
p(s+1,a%m)
``` | instruction | 0 | 40,303 | 22 | 80,606 |
Yes | output | 1 | 40,303 | 22 | 80,607 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
q=int(input())
m=10**9+7
for _ in range(q):
x,y=sorted(map(int,input().split()))
s=0
i=j=1
while j<=x and i+j<=y:
i,j=j,i+j
s+=1
if x==1 or y<3:
print(1,x*y%m)
else:
a=max(0,(y-j)//i+1)+max(0,(x-j)//i+1)
i=j=1
for c in range(s-1):
k,l=j,i+j*2
for _ in range(s-c-1):
k,l=l,k+l
if x>=k:
a+=max(0,(y-l)//k+1)
if y>=k:
a+=max(0,(x-l)//k+1)
i,j=j,i+j
print(s,a % m)
``` | instruction | 0 | 40,304 | 22 | 80,608 |
Yes | output | 1 | 40,304 | 22 | 80,609 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
m=10**9+7
f=max
for _ in range(int(input())):
x,y=sorted(map(int,input().split()))
s=-1
i=j=1
while j<=x and i+j<=y:
i,j=j,i+j
s+=1
if x==1 or y<3:print(1,x*y%m)
else:
a=max(0,(y-j)//i+1)+f(0,(x-j)//i+1)
i=j=1
for c in range(s):
k,l=j,i+j*2
for _ in range(s-c):k,l=l,k+l
if x>=k:a+=f(0,(y-l)//k+1)
if y>=k:a+=f(0,(x-l)//k+1)
i,j=j,i+j
print(s+1,a%m)
``` | instruction | 0 | 40,305 | 22 | 80,610 |
Yes | output | 1 | 40,305 | 22 | 80,611 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
F=[0,1,1]
for i in range(100):
F.append(F[-1]+F[-2])
P=0
X,Y=0,0
def f(a,b,c):
if (a*F[c]+b*F[c+1]>X or a*F[c+2]+b*F[c+1]>Y)and(a*F[c]+b*F[c+1]>Y or a*F[c+2]+b*F[c+1]>X):
return True
global P
if c:
d=a+b
while d<=X:
if f(b,d,c-1):
break
d+=b
else:
d=a+b
if b<=X and d<=Y:
P+=(Y-d)//b+1
if d<=X and b<=Y:
P+=(Y-b)//d+1
return False
mod=10**9+7
for i in range(int(input())):
P=0
X,Y=map(int,input().split())
if X==1 and Y==1:
print('1 1')
continue
if X>Y:
X,Y=Y,X
for j in range(100):
if F[j+1]>Y:
M=j-2
print(M,end=' ')
break
if F[j]>X:
M=j-2
print(M,end=' ')
break
f(0,1,M)
if M==1:
print(X*Y%mod)
else:
print(P%mod)
``` | instruction | 0 | 40,306 | 22 | 80,612 |
No | output | 1 | 40,306 | 22 | 80,613 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
from math import gcd
q = int(input())
mod = 10**9+7
for _ in range(q):
x, y = sorted(map(int, input().split()))
ma = 0
i, j = 1, 1
while j <= x and i+j <= y:
i, j = j, i+j
ma += 1
if x == 1:
print(1, x*y % mod)
else:
ans = 0
if x >= i:
ans += max(0, (y-j) // i+1)
if y >= i:
ans += max(0, (x-j) // i+1)
i, j = 1, 1
for c in range(ma-1):
k, l = j, i+j*2
if gcd(k, l) == 1:
for _ in range(ma-c-1):
k, l = l, k+l
if x >= k:
ans += max(0, (y-l) // k+1)
if y >= k:
ans += max(0, (x-l) // k+1)
i, j = j, i+j
print(ma, ans % mod)
``` | instruction | 0 | 40,307 | 22 | 80,614 |
No | output | 1 | 40,307 | 22 | 80,615 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
from math import gcd
q = int(input())
mod = 10**9+7
for _ in range(q):
x, y = sorted(map(int, input().split()))
ma = 0
i, j = 1, 1
while j <= x and i+j <= y:
i, j = j, i+j
ma += 1
if x == 1 or y<=2:
print(1, x*y % mod)
else:
ans = 0
if x >= i:
ans += max(0, (y-j) // i+1)
if y >= i:
ans += max(0, (x-j) // i+1)
i, j = 1, 1
for c in range(ma-1):
k, l = j, i+j*2
if gcd(k, l) == 1:
for _ in range(ma-c-1):
k, l = l, k+l
if x >= k:
ans += max(0, (y-l) // k+1)
if y >= k:
ans += max(0, (x-l) // k+1)
i, j = j, i+j
print(ma, ans % mod)
``` | instruction | 0 | 40,308 | 22 | 80,616 |
No | output | 1 | 40,308 | 22 | 80,617 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Kenus, the organizer of International Euclidean Olympiad, is seeking a pair of two integers that requires many steps to find its greatest common divisor using the Euclidean algorithm.
You are given Q queries. The i-th query is represented as a pair of two integers X_i and Y_i, and asks you the following: among all pairs of two integers (x,y) such that 1 ≤ x ≤ X_i and 1 ≤ y ≤ Y_i, find the maximum Euclidean step count (defined below), and how many pairs have the maximum step count, modulo 10^9+7.
Process all the queries. Here, the Euclidean step count of a pair of two non-negative integers (a,b) is defined as follows:
* (a,b) and (b,a) have the same Euclidean step count.
* (0,a) has a Euclidean step count of 0.
* If a > 0 and a ≤ b, let p and q be a unique pair of integers such that b=pa+q and 0 ≤ q < a. Then, the Euclidean step count of (a,b) is the Euclidean step count of (q,a) plus 1.
Constraints
* 1 ≤ Q ≤ 3 × 10^5
* 1 ≤ X_i,Y_i ≤ 10^{18}(1 ≤ i ≤ Q)
Input
The input is given from Standard Input in the following format:
Q
X_1 Y_1
:
X_Q Y_Q
Output
For each query, print the maximum Euclidean step count, and the number of the pairs that have the maximum step count, modulo 10^9+7, with a space in between.
Examples
Input
3
4 4
6 10
12 11
Output
2 4
4 1
4 7
Input
10
1 1
2 2
5 1000000000000000000
7 3
1 334334334334334334
23847657 23458792534
111111111 111111111
7 7
4 19
9 10
Output
1 1
1 4
4 600000013
3 1
1 993994017
35 37447
38 2
3 6
3 9
4 2
Submitted Solution:
```
from math import gcd
q = int(input())
mod = 10**9+7
for _ in range(q):
x, y = sorted(map(int, input().split()))
ma = 0
i, j = 1, 1
while j <= x and i+j <= y:
i, j = j, i+j
ma += 1
if x == 1:
print(ma, x*y % mod)
else:
ans = 0
if x >= i:
ans += max(0, (y-j) // i+1)
if y >= i:
ans += max(0, (x-j) // i+1)
i, j = 1, 1
for c in range(ma-1):
k, l = j, i+j*2
if gcd(k, l) == 1:
for _ in range(ma-c-1):
k, l = l, k+l
if x >= k:
ans += max(0, (y-l) // k+1)
if y >= k:
ans += max(0, (x-l) // k+1)
i, j = j, i+j
print(ma, ans % mod)
``` | instruction | 0 | 40,309 | 22 | 80,618 |
No | output | 1 | 40,309 | 22 | 80,619 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
t=int(input())
while t:
t-=1
n, k=list(map(int,input().split()))
a=list(map(int,input().split()))
s=set()
s.add(0)
d=dict()
for i in a:
x=k-(i%k)
if x==k:
continue
else:
y=d.get(x,x)
s.add(y)
d[x]=y+k
if len(s)==1:
print(0)
else:
print(max(s)+1)
``` | instruction | 0 | 40,599 | 22 | 81,198 |
Yes | output | 1 | 40,599 | 22 | 81,199 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
import atexit
import io
import sys
import math
from collections import defaultdict,Counter
# _INPUT_LINES = sys.stdin.read().splitlines()
# input = iter(_INPUT_LINES).__next__
# _OUTPUT_BUFFER = io.StringIO()
# sys.stdout = _OUTPUT_BUFFER
# @atexit.register
# def write():
# sys.__stdout__.write(_OUTPUT_BUFFER.getvalue())
# sys.stdout=open("CP3/output.txt",'w')
# sys.stdin=open("CP3/input.txt",'r')
# m=pow(10,9)+7
t=int(input())
for i in range(t):
n,k=map(int,input().split())
l=[-1]*n
a=list(map(int,input().split()))
s=set()
d1={}
for j in range(n):
if a[j]%k!=0:
d=k-(a[j]%k)
if d in s:
d1[d]+=1
d+=(d1[d]-1)*k
l[j]=d
s.add(d)
s1=d1.setdefault(d,1)
print(max(l)+1)
``` | instruction | 0 | 40,600 | 22 | 81,200 |
Yes | output | 1 | 40,600 | 22 | 81,201 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
from sys import stdin
inp = lambda : stdin.readline().strip()
t = int(inp())
for _ in range(t):
n, k = [int(x) for x in inp().split()]
a = [int(x) for x in inp().split()]
ar = []
for i in a:
if i%k:
ar.append(i)
ar.sort(key = lambda x: x%k)
rem = dict()
for i in ar:
if rem.get(i%k,-1) != -1:
rem[i%k] += 1
else:
rem[i%k] = 1
ans = 0
maximum = 0
for i in rem.keys():
if rem[i] > maximum:
ans = 0
ans +=1
ans += k-i
ans += (rem[i]-1)*k
maximum = rem[i]
print(ans)
``` | instruction | 0 | 40,601 | 22 | 81,202 |
Yes | output | 1 | 40,601 | 22 | 81,203 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
t=int(input())
for i in range(t):
n,k=map(int,input().split())
ar=[int(i)for i in input().split()]
for x in range(n):
ar[x]=(ar[x])%k
ar.sort()
freq = {}
check=0
for item in ar:
if item==0:
pass
elif (item in freq):
freq[item] += 1
else:
freq[item] = 1
check+=1
if check==0:
print(0)
else:
Keymax = max(freq, key=freq.get)
print(k*(ar.count(Keymax)-1)+k-Keymax+1)
``` | instruction | 0 | 40,602 | 22 | 81,204 |
Yes | output | 1 | 40,602 | 22 | 81,205 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
t = int(input())
for i in range(t):
n,k = map(int,input().split())
A = list(map(int,input().split()))
B = {}
for j in A:
if j % k not in B:
B[j % k] = 0
B[j % k] += 1
max = 0
c = k + 1
for j in B:
if j != 0 and B[j] >= max and j <= c:
max = B[j]
c = j
if c == k + 1:
print(0)
else:
print(max * k - c + 1)
``` | instruction | 0 | 40,603 | 22 | 81,206 |
No | output | 1 | 40,603 | 22 | 81,207 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
from collections import defaultdict as dd
t=int(input())
for _ in range(t):
n,k=map(int,input().split())
a=list(map(int,input().split()))
amod=[]
d=dd(int)
ct=0
mx=0
for i in range(n):
amod.append((k-(a[i]%k))%k)
d[amod[i]]+=1
if amod[i]!=0:
mx=max(mx,d[amod[i]])
ct+=(max(0,(mx-1))*k)
ct+=max(amod)
if mx==0:
print(0)
else:
print(ct+1)
``` | instruction | 0 | 40,604 | 22 | 81,208 |
No | output | 1 | 40,604 | 22 | 81,209 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
t = int(input())
for tc in range(t):
n, k = map(int, input().split()); x = 0
arr = [int(z) for z in input().split()]
needed = []
cnt = 0
usages = {}
for i in range(n):
elem = arr[i]
if elem % k == 0:
d = 0
continue
else:
d = k - (elem % k)
if not usages.get(d):
usages[d] = 1
else:
usages[d] += 1
for i, j in sorted(usages.items()):
cnt = max(cnt, j)
res = 0
for i, j in sorted(usages.items()):
if j == cnt:
res = k * (j-1) + i
print(res+1)
``` | instruction | 0 | 40,605 | 22 | 81,210 |
No | output | 1 | 40,605 | 22 | 81,211 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given an array a consisting of n positive integers.
Initially, you have an integer x = 0. During one move, you can do one of the following two operations:
1. Choose exactly one i from 1 to n and increase a_i by x (a_i := a_i + x), then increase x by 1 (x := x + 1).
2. Just increase x by 1 (x := x + 1).
The first operation can be applied no more than once to each i from 1 to n.
Your task is to find the minimum number of moves required to obtain such an array that each its element is divisible by k (the value k is given).
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.
The first line of the test case contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ 10^9) — the length of a and the required divisior. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the i-th element of a.
It is guaranteed that the sum of n does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5).
Output
For each test case, print the answer — the minimum number of moves required to obtain such an array that each its element is divisible by k.
Example
Input
5
4 3
1 2 1 3
10 6
8 7 1 8 3 7 5 10 8 9
5 10
20 100 50 20 100500
10 25
24 24 24 24 24 24 24 24 24 24
8 8
1 2 3 4 5 6 7 8
Output
6
18
0
227
8
Note
Consider the first test case of the example:
1. x=0, a = [1, 2, 1, 3]. Just increase x;
2. x=1, a = [1, 2, 1, 3]. Add x to the second element and increase x;
3. x=2, a = [1, 3, 1, 3]. Add x to the third element and increase x;
4. x=3, a = [1, 3, 3, 3]. Add x to the fourth element and increase x;
5. x=4, a = [1, 3, 3, 6]. Just increase x;
6. x=5, a = [1, 3, 3, 6]. Add x to the first element and increase x;
7. x=6, a = [6, 3, 3, 6]. We obtained the required array.
Note that you can't add x to the same element more than once.
Submitted Solution:
```
import sys
input=sys.stdin.readline
f=lambda :list(map(int, input().split()))
res=[]
for _ in range(int(input())):
n, k=f()
inp=f()
cnt={}
for i in range(n):
if inp[i]%k==0:
inp[i]=0
else:
inp[i]=(inp[i]//k+1)*k-inp[i]
cnt[inp[i]]=cnt.get(inp[i], 0)+1
inp.sort()
ans=0 if inp[n-1]==0 else 1
done={}
for i in range(n):
if inp[i]:
if not done.get(inp[i], False):
done[inp[i]]=True
else:
cur=inp[i]
l=1; r=cnt[cur]+1
ret=0
while l<=r:
m=(l+r)//2
if done.get(cur+m*k, False):
l=m+1
else:
ret=m
r=m-1
cur+=k*ret
# while done.get(cur, False):
# cur+=k
inp[i]=cur
done[cur]=True
res.append(ans+inp[n-1])
print('\n'.join(map(str, res)))
``` | instruction | 0 | 40,606 | 22 | 81,212 |
No | output | 1 | 40,606 | 22 | 81,213 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,724 | 22 | 81,448 |
Tags: implementation, math, number theory
Correct Solution:
```
def prim_roots(number):
if number == 2:
return 1
elif number == 4:
return 2
z, h = 1, 2
p = list()
while h < number:
k = pow(h, z, number)
while k > 1:
z += 1
k = (k * h) % number
if z == number - 1:
p.append(h)
z = 1
h += 1
return len(p)
print(prim_roots(int(input())))
``` | output | 1 | 40,724 | 22 | 81,449 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,725 | 22 | 81,450 |
Tags: implementation, math, number theory
Correct Solution:
```
def isPrimitiveRoot(x,p):
i = 2
m = phi = p - 1
while i*i <= m:
if m%i == 0:
if pow(x,phi//i,p) == 1:
return 0
while m%i == 0: m //= i
i += 1
if m > 1: return pow(x,phi//m,p) != 1
return pow(x,phi,p) == 1
p = int(input())
ans = 0
for g in range(1,p):
if isPrimitiveRoot(g,p):
ans += 1
print(ans)
``` | output | 1 | 40,725 | 22 | 81,451 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,726 | 22 | 81,452 |
Tags: implementation, math, number theory
Correct Solution:
```
p = int(input())
ans = 0
for i in range(1, p):
result = True
temp = i
for j in range(1, p):
if j == p-1:
if (temp-1) % p != 0:
result = False
else:
if (temp-1) % p == 0:
result = False
break
temp *= i
temp %= p
if result:
ans += 1
print(ans)
``` | output | 1 | 40,726 | 22 | 81,453 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,727 | 22 | 81,454 |
Tags: implementation, math, number theory
Correct Solution:
```
# python 3
"""
Modular arithmetic:
This has something to do with Extended Euclidean Algorithm
"""
def cows_and_primitive_roots(p_int):
if p_int == 2:
return 1
count = 0
for x in range(2, p_int):
x_to_i_mod_p = x
divisible = False
for i in range(2, p_int-1):
x_to_i_mod_p = (x * x_to_i_mod_p) % p_int
if x_to_i_mod_p == 1:
divisible = True
break
if not divisible:
count += 1
# print(x)
return count
if __name__ == "__main__":
"""
Inside of this is the test.
Outside is the API
"""
p = int(input())
print(cows_and_primitive_roots(p))
``` | output | 1 | 40,727 | 22 | 81,455 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,728 | 22 | 81,456 |
Tags: implementation, math, number theory
Correct Solution:
```
p=int(input())
count=0
for i in range(1,p):
z,x=1,1
for j in range(1,p-1):
x*=i
x%=p
if x==1:
z=0
break
if z==1:
x*=i
x%=p
if x==1:
count+=1
print(count)
``` | output | 1 | 40,728 | 22 | 81,457 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The cows have just learned what a primitive root is! Given a prime p, a primitive root <image> is an integer x (1 ≤ x < p) such that none of integers x - 1, x2 - 1, ..., xp - 2 - 1 are divisible by p, but xp - 1 - 1 is.
Unfortunately, computing primitive roots can be time consuming, so the cows need your help. Given a prime p, help the cows find the number of primitive roots <image>.
Input
The input contains a single line containing an integer p (2 ≤ p < 2000). It is guaranteed that p is a prime.
Output
Output on a single line the number of primitive roots <image>.
Examples
Input
3
Output
1
Input
5
Output
2
Note
The only primitive root <image> is 2.
The primitive roots <image> are 2 and 3. | instruction | 0 | 40,729 | 22 | 81,458 |
Tags: implementation, math, number theory
Correct Solution:
```
p = int(input()) - 1
def hcf(a, b):
if a is 0:
return b
else:
return hcf(b % a, a)
cnt = 0
for i in range(1, p + 1):
if hcf(i, p) is 1:
cnt += 1
print(cnt)
``` | output | 1 | 40,729 | 22 | 81,459 |
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