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 |
|---|---|---|---|---|---|
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
for s in[*open(0)][2::2]:
a=s.split();n=len(a);print(n//2)
while n>1:print(n,n-1,min(a[n-2:n],key=int),10**9+7);n-=2
``` | instruction | 0 | 30,327 | 22 | 60,654 |
Yes | output | 1 | 30,327 | 22 | 60,655 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
import time,math as mt,bisect as bs,sys
from sys import stdin,stdout
from collections import deque
from fractions import Fraction
from collections import Counter
from collections import OrderedDict
pi=3.14159265358979323846264338327950
def II(): # to take integer input
return int(stdin.readline())
def IP(): # to take tuple as input
return map(int,stdin.readline().split())
def L(): # to take list as input
return list(map(int,stdin.readline().split()))
def P(x): # to print integer,list,string etc..
return stdout.write(str(x)+"\n")
def PI(x,y): # to print tuple separatedly
return stdout.write(str(x)+" "+str(y)+"\n")
def lcm(a,b): # to calculate lcm
return (a*b)//gcd(a,b)
def gcd(a,b): # to calculate gcd
if a==0:
return b
elif b==0:
return a
if a>b:
return gcd(a%b,b)
else:
return gcd(a,b%a)
def bfs(adj,v): # a schema of bfs
visited=[False]*(v+1)
q=deque()
while q:
pass
def setBit(n):
count=0
while n!=0:
n=n&(n-1)
count+=1
return count
def readTree(n,e): # to read tree
adj=[set() for i in range(n+1)]
for i in range(e):
u1,u2=IP()
adj[u1].add(u2)
return adj
def sieve():
li=[True]*(10**3+5)
li[0],li[1]=False,False
for i in range(2,len(li),1):
if li[i]==True:
for j in range(i*i,len(li),i):
li[j]=False
prime,cur=[0]*200,0
for i in range(10**3+5):
if li[i]==True:
prime[cur]=i
cur+=1
return prime
def SPF():
mx=(10**7+1)
spf=[mx]*(mx)
spf[1]=1
for i in range(2,mx):
if spf[i]==mx:
spf[i]=i
for j in range(i*i,mx,i):
if i<spf[j]:
spf[j]=i
return spf
def prime(n,d):
while n!=1:
d[spf[n]]=d.get(spf[n],0)+1
n=n//spf[n]
return
#####################################################################################
mod = 1000000007
inf = 1e18
def solve():
n=II()
a=L()
mn = inf
for i in range(n):
if a[i]<mn:
mn=a[i]
note=i
print(n-1)
for i in range(n):
if i!=note:
if abs((note-i))%2==0:
print(i+1,note+1,mn,mn)
else:
print(i+1,note+1,mn+1,mn)
return
t=II()
for i in range(t):
solve()
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``` | instruction | 0 | 30,328 | 22 | 60,656 |
Yes | output | 1 | 30,328 | 22 | 60,657 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
from sys import stdin, stdout
from math import floor, gcd, fabs, factorial, fmod, sqrt, inf, log
from collections import defaultdict as dd, deque
from heapq import merge, heapify, heappop, heappush, nsmallest
from bisect import bisect_left as bl, bisect_right as br, bisect
mod = pow(10, 9) + 7
mod2 = 998244353
def inp(): return stdin.readline().strip()
def iinp(): return int(inp())
def out(var, end="\n"): stdout.write(str(var)+"\n")
def outa(*var, end="\n"): stdout.write(' '.join(map(str, var)) + end)
def lmp(): return list(mp())
def mp(): return map(int, inp().split())
def smp(): return map(str, inp().split())
def l1d(n, val=0): return [val for i in range(n)]
def l2d(n, m, val=0): return [l1d(m, val) for j in range(n)]
def remadd(x, y): return 1 if x%y else 0
def ceil(a,b): return (a+b-1)//b
S1 = 'abcdefghijklmnopqrstuvwxyz'
S2 = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'
def isprime(x):
if x<=1: return False
if x in (2, 3): return True
if x%2 == 0: return False
for i in range(3, int(sqrt(x))+1, 2):
if x%i == 0: return False
return True
def gcd(a, b):
if b==0: return a
return gcd(b, a%b)
for _ in range(int(inp())):
n = iinp()
arr = lmp()
ansl = []
for i in range(n-1):
a, b = arr[i], arr[i+1]
if gcd(a, b) != 1:
t = min(a, b)
if i==0:
ansl.append((1, 2, t, t+1))
arr[i], arr[i+1] = t, t+1
elif gcd(t, arr[i-1]) != 1:
ansl.append((i+1, i+2, t+1, t))
arr[i], arr[i+1] = t+1, t
else:
ansl.append((i+1, i+2, t, t+1))
arr[i], arr[i+1] = t, t+1
print(len(ansl))
for i in ansl:
print(*i)
``` | instruction | 0 | 30,329 | 22 | 60,658 |
No | output | 1 | 30,329 | 22 | 60,659 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
from sys import stdin, stdout
import heapq
import cProfile
from collections import Counter, defaultdict, deque
from functools import reduce
import math
import threading
import sys
import time
def get_int(): return int(stdin.readline().strip())
def get_tuple(): return map(int, stdin.readline().split())
def get_list(): return list(map(int, stdin.readline().split()))
def solve():
n = get_int()
ls = get_list()
ans = []
for i in range(1,n):
if ls[i-1]>ls[i]:
temp = ls[i] +1
while i>=2 and math.gcd(temp,ls[i-2])>1:
temp += 1
ls[i-1] = temp
else:
ls[i] = ls[i-1]+1
ans.append([i,i+1,ls[i-1],ls[i]])
print(len(ans))
for row in ans:
print(*row)
testcases = True
if testcases:
t = get_int()
for _ in range(t):
solve()
else:
solve()
``` | instruction | 0 | 30,330 | 22 | 60,660 |
No | output | 1 | 30,330 | 22 | 60,661 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
from math import sqrt
#primes = [547,1993,3643,6037]
def isPrime(n):
# Corner cases
if(n <= 1):
return False
if(n <= 3):
return True
if(n % 2 == 0 or n % 3 == 0):
return False
for i in range(5,int(sqrt(n) + 1), 6):
if(n % i == 0 or n % (i + 2) == 0):
return False
return True
def nextPrime(N):
# Base case
if (N <= 1):
return 2
prime = N
found = False
while(not found):
prime = prime + 1
if(isPrime(prime) == True):
found = True
return prime
primes = [7001,7013,7727,7741]
for _ in range(int(input())):
n = int(input())
a = list(map(int, input().split()))
print(n-1)
trick = 0
ct = 1
for i in range(1, n):
print(i, i+1, end = " ")
if a[i-1]>=a[i]:
op = nextPrime(max(a[i], a[i-1]))
print( op, min(a[i], a[i-1]))
a[i-1] = op
else:
op = nextPrime(max(a[i], a[i-1]))
print(min(a[i], a[i-1]), op)
a[i] = op
``` | instruction | 0 | 30,331 | 22 | 60,662 |
No | output | 1 | 30,331 | 22 | 60,663 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Nastia has received an array of n positive integers as a gift.
She calls such an array a good that for all i (2 ≤ i ≤ n) takes place gcd(a_{i - 1}, a_{i}) = 1, where gcd(u, v) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers u and v.
You can perform the operation: select two different indices i, j (1 ≤ i, j ≤ n, i ≠ j) and two integers x, y (1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)}. Then change a_i to x and a_j to y.
The girl asks you to make the array good using at most n operations.
It can be proven that this is always possible.
Input
The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases.
The first line of each test case contains a single integer n (1 ≤ n ≤ 10^5) — the length of the array.
The second line of each test case contains n integers a_1, a_2, …, a_{n} (1 ≤ a_i ≤ 10^9) — the array which Nastia has received as a gift.
It's guaranteed that the sum of n in one test doesn't exceed 2 ⋅ 10^5.
Output
For each of t test cases print a single integer k (0 ≤ k ≤ n) — the number of operations. You don't need to minimize this number.
In each of the next k lines print 4 integers i, j, x, y (1 ≤ i ≠ j ≤ n, 1 ≤ x, y ≤ 2 ⋅ 10^9) so that min{(a_i, a_j)} = min{(x, y)} — in this manner you replace a_i with x and a_j with y.
If there are multiple answers, print any.
Example
Input
2
5
9 6 3 11 15
3
7 5 13
Output
2
1 5 11 9
2 5 7 6
0
Note
Consider the first test case.
Initially a = [9, 6, 3, 11, 15].
In the first operation replace a_1 with 11 and a_5 with 9. It's valid, because min{(a_1, a_5)} = min{(11, 9)} = 9.
After this a = [11, 6, 3, 11, 9].
In the second operation replace a_2 with 7 and a_5 with 6. It's valid, because min{(a_2, a_5)} = min{(7, 6)} = 6.
After this a = [11, 7, 3, 11, 6] — a good array.
In the second test case, the initial array is already good.
Submitted Solution:
```
import math
for _ in range(int(input())):
n=int(input())
a=list(map(int,input().split()))
e=[4,3,7,11]
f=[2,3]
if n==1:
print(1)
else:
if n%2==0:
print(n//2)
i=0
while i<n:
if a[i]>a[i+1]:
print(i+1,i+2,e[0],e[1])
else:
print(i+1,i+2,f[0],f[1])
i+=2
else:
print((n//2)+1)
i=0
while i<n-1:
if a[i]>a[i+1]:
print(i+1,i+2,e[0],e[1])
else:
print(i+1,i+2,f[0],f[1])
i+=2
if a[0]>a[-1]:
print(1,n,e[3],e[2])
else:
print(1,n,e[2],e[3])
``` | instruction | 0 | 30,332 | 22 | 60,664 |
No | output | 1 | 30,332 | 22 | 60,665 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,353 | 22 | 60,706 |
Tags: math
Correct Solution:
```
import math
def compute_gcd(x, y):
while(y):
x, y = y, x % y
return x
def solve():
p, q = list(map(int, input().split()))
p_components = list(map(int, input().split()))
q_components = list(map(int, input().split()))
if p > q:
if p > 0 and q > 0:
print("Infinity") if p_components[0] * q_components[0] > 0 else print("-Infinity")
else:
print("Infinity") if p_components[0] > 0 else print("-Infinity")
elif p == q:
gcd = compute_gcd(p_components[0], q_components[0])
print("{}/{}".format(p_components[0] // gcd, q_components[0] // gcd))
else:
print("0/1")
solve()
``` | output | 1 | 30,353 | 22 | 60,707 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,354 | 22 | 60,708 |
Tags: math
Correct Solution:
```
n,m = [int(i) for i in input().split()]
p = [int(i) for i in input().split()]
q = [int(i) for i in input().split()]
def gcd(a,b):
if(b == 0):
return a
return gcd(b,a%b)
if(len(p) < len(q)):
print('0/1')
elif(len(p) == len(q)):
a,b = p[0],q[0]
g = gcd(a,b)
print('{}/{}'.format(a//g,b//g))
elif(p[0]*q[0] > 1):
print('Infinity')
else:
print('-Infinity')
``` | output | 1 | 30,354 | 22 | 60,709 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,355 | 22 | 60,710 |
Tags: math
Correct Solution:
```
import sys
input = sys.stdin.readline
n,m = map(int,input().split())
arr = list(map(int,input().split()))
brr = list(map(int,input().split()))
if n < m:
print("0/1")
elif n > m:
if (arr[0] < 0 and brr[0] < 0) or (arr[0] > 0 and brr[0] > 0):
print("Infinity")
else:
print("-Infinity")
else:
from fractions import Fraction
num = Fraction(arr[0],brr[0])
print(str(num.numerator) + '/' + str(num.denominator))
``` | output | 1 | 30,355 | 22 | 60,711 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,356 | 22 | 60,712 |
Tags: math
Correct Solution:
```
import math
n,m = map(int, input().strip().split(' '))
a = list(map(int, input().strip().split(' ')))
b = list(map(int, input().strip().split(' ')))
if n==m:
p1=math.gcd(a[0],b[0])
a[0]=a[0]//p1
b[0]=b[0]//p1
if a[0]*b[0]>0:
print(str(abs(a[0]))+'/'+str(abs(b[0])))
else:
print('-'+str(abs(a[0]))+'/'+str(abs(b[0])))
elif m>n:
print('0/1')
else:
if a[0]*b[0]>0:
print('Infinity')
else:
print('-Infinity')
``` | output | 1 | 30,356 | 22 | 60,713 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,357 | 22 | 60,714 |
Tags: math
Correct Solution:
```
def f(a,b):
if a%b==0:
return b
return f(b,a%b)
a,b=map(int,input().split())
l1=list(map(int,input().split()))
l2=list(map(int,input().split()))
if len(l1)>len(l2):
x='Infinity'
if (l1[0]>0 and l2[0]>0) or (l1[0]<0 and l2[0]<0) :
print('Infinity')
else:
print('-Infinity')
elif len(l1)<len(l2):
print('0/1',sep='')
else:
n=l1[0];m=l2[0]
g=f(n,m)
print(n//g,'/',m//g,sep='')
``` | output | 1 | 30,357 | 22 | 60,715 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,358 | 22 | 60,716 |
Tags: math
Correct Solution:
```
from math import gcd
n, m = map(int, input().split())
a = int(input().split()[0])
b = int(input().split()[0])
if n == m:
g = gcd(a, b)
a //= g
b //= g
if b < 0:
a = -a
b = -b
print('{}/{}'.format(a, b))
elif n > m:
if (a > 0) == (b > 0):
print('Infinity')
else:
print('-Infinity')
else:
print('0/1')
``` | output | 1 | 30,358 | 22 | 60,717 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,359 | 22 | 60,718 |
Tags: math
Correct Solution:
```
from fractions import Fraction
def limit(a,b):
if len(a)>len(b):
if (a[0]>0 and b[0]>0) or (a[0]<0 and b[0]<0):
return "Infinity"
else:
return "-Infinity"
if len(b)>len(a):
return "0"+"/"+"1"
else:
c=""
if a[0]<0 and b[0]>0:
c="-"
if a[0]>0 and b[0]<0:
c="-"
a[0]=abs(a[0])
b[0]=abs(b[0])
ans=Fraction(a[0] / b[0]).limit_denominator().as_integer_ratio()
return c+str(ans[0])+"/"+str(ans[1])
a=input()
lst=list(map(int,input().strip().split()))
lst2=list(map(int,input().strip().split()))
print(limit(lst,lst2))
``` | output | 1 | 30,359 | 22 | 60,719 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function | instruction | 0 | 30,360 | 22 | 60,720 |
Tags: math
Correct Solution:
```
def hcfnaive(a,b):
if(b==0):
return a
else:
return hcfnaive(b,a%b)
n,m=list(map(int,input().split()))
a=list(map(int,input().split()))
b=list(map(int,input().split()))
if n>m:
if a[0]/b[0]>0:
print("Infinity")
else:
print( "-Infinity")
elif n<m:
print('0/1')
else:
p=a[0]
q=b[0]
s=''
if p/q<0:
s='-'
p=abs(p)
q=abs(q)
h=hcfnaive(p,q)
p=p//h
q=q//h
print(s+str(p)+'/'+str(q))
``` | output | 1 | 30,360 | 22 | 60,721 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two polynomials:
* P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and
* Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm.
Calculate limit <image>.
Input
The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly.
The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0).
The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0).
Output
If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes).
If the value of the limit equals zero, print "0/1" (without the quotes).
Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction.
Examples
Input
2 1
1 1 1
2 5
Output
Infinity
Input
1 0
-1 3
2
Output
-Infinity
Input
0 1
1
1 0
Output
0/1
Input
2 2
2 1 6
4 5 -7
Output
1/2
Input
1 1
9 0
-5 2
Output
-9/5
Note
Let's consider all samples:
1. <image>
2. <image>
3. <image>
4. <image>
5. <image>
You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function
Submitted Solution:
```
from math import gcd
n, m = map(int, input().split())
p = list(map(int, input().split()))
q = list(map(int, input().split()))
if (n>m):
if (p[0]<0 and q[0]>0) or (p[0]>0 and q[0]<0):
print("-Infinity")
else:
print("Infinity")
elif (n==m):
d = gcd(p[0], q[0])
x = p[0] // d
y = q[0] // d
if (x<0 and y>0) or (x>0 and y<0):
print("-" + str(abs(x)) + "/" + str(abs(y)))
else:
print(str(abs(x)) + "/" + str(abs(y)))
else:
print("0/1")
``` | instruction | 0 | 30,364 | 22 | 60,728 |
Yes | output | 1 | 30,364 | 22 | 60,729 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,633 | 22 | 61,266 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
def bin_pow(num, degree , module):
if degree == 0:
return 1
if degree == 1:
return num % module
if degree % 2 == 0:
val = bin_pow(num, degree // 2, module)
return (val * val) % module
return (num * bin_pow(num , degree - 1, module)) % module
x, y = map(int,input().split())
if y % x != 0:
print(0)
exit(0)
y //= x
divs = set()
to_gen = []
num = 2
val = y
while num * num <= val:
degree = 0
while y % num == 0:
degree+=1
y //= num
if degree != 0:
to_gen.append((num, degree))
if num == 2:
num += 1
else:
num += 2
if y != 1:
to_gen.append((y, 1))
to_gen_len = len(to_gen)
def generate(ind):
if ind == to_gen_len:
yield 1
return
gen_val = to_gen[ind][0]
for deg in range(1 + to_gen[ind][1]):
for each in generate(ind + 1):
yield gen_val**deg * each
for each in generate(0):
divs.add(each)
divs = list(divs)
divs.sort()
divs_answers = {}
mod = 10**9 + 7
ans = bin_pow(2, val - 1, mod)
for el in divs:
if el == 1:
divs_answers[el] = 1
ans -= 1
else:
curr_val = bin_pow(2, el - 1 ,mod)
for other_el in divs:
if other_el >= el:
break
if el % other_el !=0:
continue
curr_val -= divs_answers[other_el]
divs_answers[el] = curr_val % mod
ans -= curr_val
print(divs_answers[val])
``` | output | 1 | 30,633 | 22 | 61,267 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,634 | 22 | 61,268 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
x, y = map(int, input().split())
b = y // x
if y % x != 0:
exit(print(0))
ds = set()
M = 10**9 + 7
for i in range(1, int(pow(b,0.5)+1)):
if b % i == 0:
ds.add(i)
ds.add(b//i)
ds = sorted(list(ds))
ans = pow(2, b-1, M)
f = ds[::]
for i in range(len(ds)):
f[i] = pow(2, ds[i]-1, M)
for j in range(i):
if ds[i] % ds[j] == 0:
f[i] -= f[j]
print(f[-1]%M)
``` | output | 1 | 30,634 | 22 | 61,269 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,635 | 22 | 61,270 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
MOD = 1000000007
x, y = map(int,input().split())
if y%x == 0: # 'y' must be divisible by 'x'
div = y // x
y_mult = set() # Distinc common multiples for y
for i in range(1, int(pow(div, 0.5) + 1)): # Just is neccesary until root of div
if div % i == 0:
y_mult.add(i)
y_mult.add(div // i)
y_mult = sorted(list(y_mult))
ym_copy = y_mult.copy()
for i in range(len(ym_copy)):
ym_copy[i] = pow(2, y_mult[i]-1, MOD) # Efficient pow
for j in range(i):
if y_mult[i]%y_mult[j] == 0:
ym_copy[i] -= ym_copy[j]
print(ym_copy[-1]%MOD)
else:
print(0)
``` | output | 1 | 30,635 | 22 | 61,271 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,636 | 22 | 61,272 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
#!/usr/bin/env python3
from fractions import gcd
from operator import mul
from functools import reduce
from itertools import combinations
eval_function = lambda x: lambda f: f(x)
@eval_function(int((10**9)**0.5))
def prime(n):
sieve = [True] * (n+1)
sieve[0] = sieve[1] = False
index = 2
for i in range(int(len(sieve)**0.5)):
if sieve[i]:
for j in range(2*i, len(sieve), i):
sieve[j] = False
index += 1
return [i for i, is_prime in enumerate(sieve) if is_prime]
def factorized(n):
factors = []
for i in prime:
if i**2 > n:
break
while not n % i:
factors += [i]
n //= i
if n > 1:
factors += [n]
return factors
def solve(x, y, mod=None):
if gcd(x, y) != x:
return 0
y = y//x
c = pow(2, y-1, mod)
unique_factors = set(factorized(y))
for i in range(1, len(unique_factors)+1):
for divisor in combinations(unique_factors, i):
d = reduce(mul, divisor)
c += (-1)**i * pow(2, y//d-1, mod)
c %= mod
return c
def main():
x, y = [int(n) for n in input().split()]
print(solve(x, y, 10**9+7))
if __name__ == '__main__':
main()
``` | output | 1 | 30,636 | 22 | 61,273 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,637 | 22 | 61,274 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
from collections import defaultdict
x, y = map(int, input().split())
if y%x != 0:
print(0)
exit()
x = y//x
divisors = []
for i in range(1,int(x**(.5))+1):
if x%i == 0:
divisors.append(i)
divisors.append(x//i)
if divisors[-1] == divisors[-2]: divisors.pop()
divisors.sort()
va = defaultdict(int)
va[1] = 1
mod = 10**9 + 7
for i in range(1, len(divisors)):
k = divisors[i]
count = pow(2,k-1,mod)
for j in range(i):
if k%divisors[j] == 0: count = (count-va[divisors[j]])%mod
va[k] = count
print(va[x])
``` | output | 1 | 30,637 | 22 | 61,275 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,638 | 22 | 61,276 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
import math
mod= 1000000007
x, y = [int(x) for x in input().split()]
if y%x != 0:
print(0)
exit()
y= y//x
seqs=set()
for x in range(1, int(math.sqrt(y) + 1)):
if y%x != 0:
continue
seqs.add(x)
seqs.add(y// x)
seqs=sorted(list(seqs))
ordered= seqs.copy()
for i in range(len(seqs)):
ordered[i]=pow(2, seqs[i] - 1, mod)
for j in range(i):
if seqs[i]% seqs[j] == 0:
ordered[i]-= ordered[j]
print(int(ordered[len(ordered)-1] % mod))
``` | output | 1 | 30,638 | 22 | 61,277 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,639 | 22 | 61,278 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
p=1000000007
def mul(x,y):
rt=1
while y>0:
if y%2==1:
rt=(rt*x)%p
x=(x*x)%p
y=y//2
return rt
x,y=map(int,input().split())
if(y%x!=0):print(0)
else:
y/=x
d=set([])
i=1
while i*i<=y:
if y%i==0:
d.add(i);d.add(y/i)
i+=1
d=sorted(list(d))
dp=d[::]
for i in range(len(d)):
dp[i]=mul(2,d[i]-1)
for j in range(i):
if d[i]%d[j]==0:
dp[i]-=dp[j]
print(dp[-1]%p)
``` | output | 1 | 30,639 | 22 | 61,279 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test. | instruction | 0 | 30,640 | 22 | 61,280 |
Tags: bitmasks, combinatorics, dp, math, number theory
Correct Solution:
```
modulo=10**9+7
num1,num2=map(int,input().split())
if num2%num1:
print("0")
else:
num2=num2//num1
arr=set()
for i in range(1,int(pow(num2,0.5)+1)):
if num2%i==0:
arr.add(i)
arr.add(num2//i)
arr=sorted(list(arr))
cop2=arr[::]
for i in range(len(cop2)):
cop2[i]=pow(2,arr[i]-1,modulo)
for j in range(i):
if arr[i]%arr[j]==0:
cop2[i]-=cop2[j]
print(cop2[-1]%modulo)
``` | output | 1 | 30,640 | 22 | 61,281 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
M=10**9+7
a,b=map(int,input().split())
if b%a:print(0)
else:
b//=a
d=set()
for i in range(1,int(pow(b,0.5)+1)):
if b%i==0:
d.add(i)
d.add(b//i)
d=sorted(list(d))
f=d[::]
for i in range(len(f)):
f[i]=pow(2,d[i]-1,M)
for j in range(i):
if d[i]%d[j]==0:
f[i]-=f[j]
print(f[-1]%M)
``` | instruction | 0 | 30,641 | 22 | 61,282 |
Yes | output | 1 | 30,641 | 22 | 61,283 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
import sys
mod = 1000000007
def mul(x,y):
res=1
while y>0:
if y%2==1:
res=(res*x)%mod
x=(x*x)%mod
y//=2
return res
x,y = map(int,sys.stdin.readline().split())
if(y%x!=0):
print(0)
else:
y/=x
d=set([])
i=1
while i*i<=y:
if y%i==0:
d.add(i);d.add(y/i)
i+=1
d=sorted(list(d))
dp=d[::]
for i in range(len(d)):
dp[i]=mul(2,d[i]-1)
for j in range(i):
if d[i]%d[j]==0:
dp[i]-=dp[j]
print(dp[-1]%mod)
``` | instruction | 0 | 30,642 | 22 | 61,284 |
Yes | output | 1 | 30,642 | 22 | 61,285 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
'''
midpow function copy marsi.
39 no line thaki 51 copy marsi.
tou kita oiche
as if i care
'''
import sys
mod = 1000000007
def modpow(a, x):
if x == 0:
return 1
if x == 1:
return a
b = modpow(a, x//2)
return (b*b*(a ** (x%2))) % mod
n,m = map(int, input().split())
#ans = gcd(n,m)
if m%n != 0:
print(0)
sys.exit(0)
if n == m:
print(1)
sys.exit(0)
ans = m//n
arr = []
i = 2
while i*i < ans:
if ans%i == 0:
arr.append(i)
arr.append(ans//i)
i += 1
if int(ans ** 0.5) ** 2 == ans:
arr.append(int(ans** 0.5))
arr.append(ans)
arr.sort()
result = modpow(2, ans -1)
muls = [0 for _ in arr]
muls[0] = 1
for xid,d in enumerate(arr):
prevs = 0
for id,d2 in enumerate(arr):
if d2 < d and d%d2 ==0:
prevs += muls[id]
muls[xid] = (1 - prevs)
result += (prevs - 1) * modpow(2, ans//d - 1)
result = (result + 1000 * mod) % mod
print(result)
``` | instruction | 0 | 30,643 | 22 | 61,286 |
Yes | output | 1 | 30,643 | 22 | 61,287 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
import operator, math, itertools, functools
mod = int(1e9 + 7)
def factor(n):
a = []
for i in range(2, int(math.sqrt(n))+1):
while n % i == 0:
a.append(i)
n //= i
if n > 1: a.append(n)
return a
def comp(n):
f = set(factor(n))
c = pow(2, n-1, mod)
for i in range(1, len(f)+1):
if i & 1:
sign = -1
else:
sign = 1
for j in itertools.combinations(f, i):
k = functools.reduce(operator.mul, j)
c += sign * pow(2, n//k-1, mod)
c = c % mod
return c
x, y = map(int, input().split())
if y % x == 0:
print(comp(y // x))
else:
print(0)
``` | instruction | 0 | 30,644 | 22 | 61,288 |
Yes | output | 1 | 30,644 | 22 | 61,289 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
lista=input().split(' ')
def Unusual(x,y):
if y%x!=0 or y<=x:
return 0
lista=[]
for i in range(y+1):
lista.append([0,0])
lista[x][0]=1
lista[x][1]=1
lista[0][0]=1
i=2
while x*i<=y:
j=1
while x*i-x*j>=0:
lista[x*i][0]+=lista[x*i-x*j][0]
if j==1:
lista[x*i][1]+=lista[x*i-x][0]
else:
lista[x*i][1]+=lista[x*i-x*j][1]
j+=1
i+=1
return lista[y][1]
print(Unusual(int(lista[0]),int(lista[1])))
``` | instruction | 0 | 30,645 | 22 | 61,290 |
No | output | 1 | 30,645 | 22 | 61,291 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
mod = 1000000009
def modpow(a, x):
if x==0:
return 1
if x==1:
return a
b = modpow(a, x//2)
return b*b*(a ** (x%2)) % mod
import sys
x,y = list(map(int, input().split()))
if y%x != 0:
print(0)
sys.exit(0)
a = y//x
divs = []
i=2
while i*i < a:
if a%i == 0:
divs.append(i)
divs.append(a//i)
i+=1
if int(a ** 0.5) **2 == a:
divs.append(int(a**0.5))
divs.append(a)
divs.sort()
#print(divs)
res = modpow(2, a-1)
for d in divs:
prevs = 0
for d2 in divs:
if d2<d and d%d2 == 0:
prevs+=1
res += (prevs-1) * modpow(2, a//d - 1)
res = (res + 1000 * mod) % mod
#print(d, res, ',', prevs)
print(res)
``` | instruction | 0 | 30,646 | 22 | 61,292 |
No | output | 1 | 30,646 | 22 | 61,293 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
m, n = list(map(int, input().split()))
if n % m != 0:
print(0)
else:
temp = n//m
result = 2**(temp-1)
for x in range(1, temp//2+1):
result -= (2**(x-1)-1)
print(result-1)
``` | instruction | 0 | 30,647 | 22 | 61,294 |
No | output | 1 | 30,647 | 22 | 61,295 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7.
gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
Input
The only line contains two positive integers x and y (1 ≤ x, y ≤ 109).
Output
Print the number of such sequences modulo 109 + 7.
Examples
Input
3 9
Output
3
Input
5 8
Output
0
Note
There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3).
There are no suitable sequences in the second test.
Submitted Solution:
```
import math
mod=int(10e9+7)
num1,num2=map(int,input().split())
if num2%num1:print(0)
else:
num2=int(num2/num1)
arr=[]
for x in range(1,int(math.sqrt(num2)+1)):
if num2%x==0:
arr.append(x)
arr.append(int(num2/x))
arr=sorted(arr)
cop=arr[::]
for x in range(len(cop)):
cop[x]=pow(2,arr[x]-1,mod)
for j in range(x):
if arr[x]%arr[j]==0:
cop[x]= cop[x] - cop[j]
if cop[-1]%mod ==0:
print ("1")
else:
print(cop[-1]%mod)
``` | instruction | 0 | 30,648 | 22 | 61,296 |
No | output | 1 | 30,648 | 22 | 61,297 |
Provide a correct Python 3 solution for this coding contest problem.
There are N people, conveniently numbered 1 through N. We want to divide them into some number of groups, under the following two conditions:
* Every group contains between A and B people, inclusive.
* Let F_i be the number of the groups containing exactly i people. Then, for all i, either F_i=0 or C≤F_i≤D holds.
Find the number of these ways to divide the people into groups. Here, two ways to divide them into groups is considered different if and only if there exists two people such that they belong to the same group in exactly one of the two ways. Since the number of these ways can be extremely large, print the count modulo 10^9+7.
Constraints
* 1≤N≤10^3
* 1≤A≤B≤N
* 1≤C≤D≤N
Input
The input is given from Standard Input in the following format:
N A B C D
Output
Print the number of ways to divide the people into groups under the conditions, modulo 10^9+7.
Examples
Input
3 1 3 1 2
Output
4
Input
7 2 3 1 3
Output
105
Input
1000 1 1000 1 1000
Output
465231251
Input
10 3 4 2 5
Output
0 | instruction | 0 | 30,816 | 22 | 61,632 |
"Correct Solution:
```
def main():
mod = 10**9+7
inv_n = [0]*1001
nCr = [[1]*(i+1) for i in range(1001)]
for i in range(1001):
inv_n[i] = pow(i, mod-2, mod)
for i in range(2, 1001):
for j in range(1, i):
nCr[i][j] = (nCr[i-1][j-1]+nCr[i-1][j]) % mod
n, a, b, c, d = map(int, input().split())
dp = [0]*(n+1)
dp[0] = 1
for A in range(b, a-1, -1):
dp2 = [i for i in dp]
for N in range(n-c*A, -1, -1):
e = dp[N]
if e:
temp = 1
for C in range(1, c):
temp = temp*nCr[n-N-(C-1)*A][A]*inv_n[C] % mod
for C in range(c, min(d, (n-N)//A)+1):
temp = temp*nCr[n-N-(C-1)*A][A]*inv_n[C] % mod
dp2[N+C*A] = (dp2[N+C*A]+temp*e) % mod
dp = dp2
print(dp[-1])
main()
``` | output | 1 | 30,816 | 22 | 61,633 |
Provide a correct Python 3 solution for this coding contest problem.
There are N people, conveniently numbered 1 through N. We want to divide them into some number of groups, under the following two conditions:
* Every group contains between A and B people, inclusive.
* Let F_i be the number of the groups containing exactly i people. Then, for all i, either F_i=0 or C≤F_i≤D holds.
Find the number of these ways to divide the people into groups. Here, two ways to divide them into groups is considered different if and only if there exists two people such that they belong to the same group in exactly one of the two ways. Since the number of these ways can be extremely large, print the count modulo 10^9+7.
Constraints
* 1≤N≤10^3
* 1≤A≤B≤N
* 1≤C≤D≤N
Input
The input is given from Standard Input in the following format:
N A B C D
Output
Print the number of ways to divide the people into groups under the conditions, modulo 10^9+7.
Examples
Input
3 1 3 1 2
Output
4
Input
7 2 3 1 3
Output
105
Input
1000 1 1000 1 1000
Output
465231251
Input
10 3 4 2 5
Output
0 | instruction | 0 | 30,818 | 22 | 61,636 |
"Correct Solution:
```
from collections import defaultdict
N,A,B,C,D=map(int,input().split())
mod=10**9+7
table=[1]*(N+3)
t=1
for i in range(1,N+3):
t*=i
t%=mod
table[i]=t
rtable=[1]*(N+3)
t=1
for i in range(1,N+3):
t*=pow(i,mod-2,mod)
t%=mod
rtable[i]=t
dp=defaultdict(int)
dp[N]=1
for l in range(A,B+1):
ndp=defaultdict(int)
for mem,num in dp.items():
ndp[mem]+=num
for k in range(C,D+1):
if mem-k*l<0:
break
t=(table[mem]*pow(rtable[l],k,mod)*rtable[mem-k*l]*rtable[k])%mod
ndp[mem-k*l]+=(num*t)%mod
ndp[mem-k*l]%mod
#print(mem-k*l,(num*t)%mod)
dp=ndp
#print(dp)
print(dp[0]%mod)
``` | output | 1 | 30,818 | 22 | 61,637 |
Provide a correct Python 3 solution for this coding contest problem.
There are N people, conveniently numbered 1 through N. We want to divide them into some number of groups, under the following two conditions:
* Every group contains between A and B people, inclusive.
* Let F_i be the number of the groups containing exactly i people. Then, for all i, either F_i=0 or C≤F_i≤D holds.
Find the number of these ways to divide the people into groups. Here, two ways to divide them into groups is considered different if and only if there exists two people such that they belong to the same group in exactly one of the two ways. Since the number of these ways can be extremely large, print the count modulo 10^9+7.
Constraints
* 1≤N≤10^3
* 1≤A≤B≤N
* 1≤C≤D≤N
Input
The input is given from Standard Input in the following format:
N A B C D
Output
Print the number of ways to divide the people into groups under the conditions, modulo 10^9+7.
Examples
Input
3 1 3 1 2
Output
4
Input
7 2 3 1 3
Output
105
Input
1000 1 1000 1 1000
Output
465231251
Input
10 3 4 2 5
Output
0 | instruction | 0 | 30,820 | 22 | 61,640 |
"Correct Solution:
```
N, A, B, C, D = map(int, input().split())
mod = 7 + 10 ** 9
fact = [1] * (N+1)
frev = [1] * (N+1)
for i in range(1, N+1):
temp = fact[i] = (fact[i-1] * i) % mod
frev[i] = pow(temp, mod-2, mod)
def P(n, r):
return (fact[n] * frev[n-r]) % mod
DP = [[0 for j in range(N+1)] for i in range(N+1)]
for i in range(N+1): DP[i][0] = 1
for i in range(A, B+1):
revi = frev[i]
for j in range(1, N+1):
dpij = 0 + DP[i-1][j]
for k in range(C, D + 1):
if j - i * k < 0: break
temp = (P(N-j+i*k, i*k) * pow(revi, k, mod) * frev[k]) % mod
dpij += (temp * DP[i-1][j-i*k]) % mod
DP[i][j] = dpij % mod
print(DP[B][N])
``` | output | 1 | 30,820 | 22 | 61,641 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,891 | 22 | 61,782 |
"Correct Solution:
```
import math
try:
while True:
a,b = input().split(" ")
print(math.gcd(int(a),int(b)))
except EOFError:
m = 0
``` | output | 1 | 30,891 | 22 | 61,783 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,892 | 22 | 61,784 |
"Correct Solution:
```
import sys
for e in sys.stdin:
p, q = map(int, e.split())
if p < q:p, q = q, p
while True:
if p % q == 0:
print(q)
break
else:
p, q = q, p % q
``` | output | 1 | 30,892 | 22 | 61,785 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,893 | 22 | 61,786 |
"Correct Solution:
```
while True:
try:
a, b= map(int, input().split())
while b!= 0:
a, b= b, a%b
print(a)
except:
break
``` | output | 1 | 30,893 | 22 | 61,787 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,894 | 22 | 61,788 |
"Correct Solution:
```
import math
while True:
try:print(math.gcd(*map(int,input().split(" "))))
except:break
``` | output | 1 | 30,894 | 22 | 61,789 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,895 | 22 | 61,790 |
"Correct Solution:
```
while 1:
try:a,b=map(int,input().split())
except:break
while b:a,b=b,a%b
print(a)
``` | output | 1 | 30,895 | 22 | 61,791 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,896 | 22 | 61,792 |
"Correct Solution:
```
def gcd(x, y):
if y==0: return x
z = x % y
return gcd(y, z)
while True:
try:
a, b = (int(i) for i in input().split())
except EOFError:
break
if a<b: print(gcd(b,a))
else: print(gcd(a,b))
``` | output | 1 | 30,896 | 22 | 61,793 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,897 | 22 | 61,794 |
"Correct Solution:
```
from math import gcd
from sys import stdin
for line in stdin:
a,b = [int(i) for i in line.split()]
print(gcd(a,b))
``` | output | 1 | 30,897 | 22 | 61,795 |
Provide a correct Python 3 solution for this coding contest problem.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12 | instruction | 0 | 30,898 | 22 | 61,796 |
"Correct Solution:
```
import math
def get_input():
while True:
try:
yield ''.join(input())
except EOFError:
break
N = list(get_input())
for l in range(len(N)):
a,b = [int(i) for i in N[l].split()]
print(math.gcd(a, b))
``` | output | 1 | 30,898 | 22 | 61,797 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12
Submitted Solution:
```
# -*- coding: utf-8 -*-
"""
"""
import sys
from sys import stdin
import fractions
input = stdin.readline
def main(args):
for line in stdin:
n, m = map(int, line.split())
print(fractions.gcd(n, m))
if __name__ == '__main__':
main(sys.argv[1:])
``` | instruction | 0 | 30,899 | 22 | 61,798 |
Yes | output | 1 | 30,899 | 22 | 61,799 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12
Submitted Solution:
```
def main():
while True:
try:
a, b = map(int, input().split())
print(euclid_algorithm(a, b))
except EOFError as e:
break
def euclid_algorithm(a, b):
while a and b:
if a > b:
a = a - b
else:
b = b - a
return a if a else b
if __name__ == '__main__':
main()
``` | instruction | 0 | 30,900 | 22 | 61,800 |
Yes | output | 1 | 30,900 | 22 | 61,801 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12
Submitted Solution:
```
import sys
def gcd(a,b):
if(b):
return gcd(b,(a%b))
else:
return a
List = []
for i in sys.stdin:
List.append(i)
for data in List:
print(gcd(int(data.split()[0]),int(data.split()[1])))
``` | instruction | 0 | 30,901 | 22 | 61,802 |
Yes | output | 1 | 30,901 | 22 | 61,803 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12
Submitted Solution:
```
# AOJ 1009: Greatest Common Divisor
# Python3 2018.7.4 bal4u
def gcd(a, b):
while b != 0:
r = a % b
a, b = b, r
return a
while True:
try: a, b = map(int, input().split())
except: break
print(gcd(a, b))
``` | instruction | 0 | 30,902 | 22 | 61,804 |
Yes | output | 1 | 30,902 | 22 | 61,805 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task.
Input
The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000.
The number of pairs (datasets) is less than 50.
Output
Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line.
Example
Input
57 38
60 84
Output
19
12
Submitted Solution:
```
from math import gcd
while True:
try:
a, b= map(int, input().split())
print(gcd(a, b))
except:
break
``` | instruction | 0 | 30,903 | 22 | 61,806 |
No | output | 1 | 30,903 | 22 | 61,807 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given three positive (i.e. strictly greater than zero) integers x, y and z.
Your task is to find positive integers a, b and c such that x = max(a, b), y = max(a, c) and z = max(b, c), or determine that it is impossible to find such a, b and c.
You have to answer t independent test cases. Print required a, b and c in any (arbitrary) order.
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 only line of the test case contains three integers x, y, and z (1 ≤ x, y, z ≤ 10^9).
Output
For each test case, print the answer:
* "NO" in the only line of the output if a solution doesn't exist;
* or "YES" in the first line and any valid triple of positive integers a, b and c (1 ≤ a, b, c ≤ 10^9) in the second line. You can print a, b and c in any order.
Example
Input
5
3 2 3
100 100 100
50 49 49
10 30 20
1 1000000000 1000000000
Output
YES
3 2 1
YES
100 100 100
NO
NO
YES
1 1 1000000000 | instruction | 0 | 31,155 | 22 | 62,310 |
Tags: math
Correct Solution:
```
for t in range(int(input())):
l=sorted(map(int,input().split()))
if(l[1]!=l[2]):
print('NO')
else:
print('YES')
print(l[0],l[0],l[2])
``` | output | 1 | 31,155 | 22 | 62,311 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given three positive (i.e. strictly greater than zero) integers x, y and z.
Your task is to find positive integers a, b and c such that x = max(a, b), y = max(a, c) and z = max(b, c), or determine that it is impossible to find such a, b and c.
You have to answer t independent test cases. Print required a, b and c in any (arbitrary) order.
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 only line of the test case contains three integers x, y, and z (1 ≤ x, y, z ≤ 10^9).
Output
For each test case, print the answer:
* "NO" in the only line of the output if a solution doesn't exist;
* or "YES" in the first line and any valid triple of positive integers a, b and c (1 ≤ a, b, c ≤ 10^9) in the second line. You can print a, b and c in any order.
Example
Input
5
3 2 3
100 100 100
50 49 49
10 30 20
1 1000000000 1000000000
Output
YES
3 2 1
YES
100 100 100
NO
NO
YES
1 1 1000000000 | instruction | 0 | 31,156 | 22 | 62,312 |
Tags: math
Correct Solution:
```
#!/usr/bin/env python3
import sys
test_cases= int(input())
list_of_cases=[]
for i in range(test_cases):
x,y,z = map(int,input().split(" "))
list_of_cases.append((x,y,z))
for i in range(test_cases):
x,y,z = list_of_cases[i]
valid_numbers_max = max(x,y,z)
a=x
b=y
flag = 0
for a in x,y,z:
for b in x,y,z:
for c in x,y,z:
#print(f"testing a,b,c as {a} {b} {c}")
if x == max(a,b) and y == max(a,c) and z == max(b,c) and flag !=1:
print("YES")
print(a,b,c)
flag = 1
break
if flag != 1:
print("NO")
``` | output | 1 | 31,156 | 22 | 62,313 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given three positive (i.e. strictly greater than zero) integers x, y and z.
Your task is to find positive integers a, b and c such that x = max(a, b), y = max(a, c) and z = max(b, c), or determine that it is impossible to find such a, b and c.
You have to answer t independent test cases. Print required a, b and c in any (arbitrary) order.
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 only line of the test case contains three integers x, y, and z (1 ≤ x, y, z ≤ 10^9).
Output
For each test case, print the answer:
* "NO" in the only line of the output if a solution doesn't exist;
* or "YES" in the first line and any valid triple of positive integers a, b and c (1 ≤ a, b, c ≤ 10^9) in the second line. You can print a, b and c in any order.
Example
Input
5
3 2 3
100 100 100
50 49 49
10 30 20
1 1000000000 1000000000
Output
YES
3 2 1
YES
100 100 100
NO
NO
YES
1 1 1000000000 | instruction | 0 | 31,157 | 22 | 62,314 |
Tags: math
Correct Solution:
```
from itertools import permutations
t = int(input())
for _ in range(t):
x,y,z=(map(int, input().split()))
l = list(set([x,y,z]))
flag = 0
if(len(l)==1):
print("YES")
print(x,y,z)
flag=1
else:
for X in range(3-len(l)):
l.append(1)
perm = list(permutations(l))
for i in (perm):
flag = 0
a = (i)[0]
b = (i)[1]
c = (i)[2]
if(x == max(a,b) and y == max(a,c) and z == max(b,c)):
print("YES")
print(i[0],i[1],i[2])
flag = 1
break
if flag == 0:
print("NO")
``` | output | 1 | 31,157 | 22 | 62,315 |
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