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.
You are given two integers a and m. Calculate the number of integers x such that 0 β€ x < m and \gcd(a, m) = \gcd(a + x, m).
Note: \gcd(a, b) is the greatest common divisor of a and b.
Input
The first line contains the single integer T (1 β€ T β€ 50) β the number of test cases.
Next T lines contain test cases β one per line. Each line contains two integers a and m (1 β€ a < m β€ 10^{10}).
Output
Print T integers β one per test case. For each test case print the number of appropriate x-s.
Example
Input
3
4 9
5 10
42 9999999967
Output
6
1
9999999966
Note
In the first test case appropriate x-s are [0, 1, 3, 4, 6, 7].
In the second test case the only appropriate x is 0.
Submitted Solution:
```
from math import sqrt, gcd
from collections import defaultdict
def primeFactors(n):
prime = defaultdict(int)
while n % 2 == 0:
prime[2] += 1
n = n//2
for i in range(3,int(sqrt(n))+1,2):
while n % i== 0:
prime[i] += 1
n = n//i
if n > 2:
prime[n] += 1
return prime
t = int(input())
for _ in range(t):
a, m = map(int, input().split())
pf = primeFactors(m)
g = gcd(a, m)
gcd_pf = primeFactors(g)
for k,v in gcd_pf.items():
pf[k] -= v
ct = 1+(a+m-1)//g - (a//g)
for k,v in pf.items():
if k != 1 and v:
ct -= (a+m-1)//(k*g) - (a//(k*g))
print(ct)
``` | instruction | 0 | 17,825 | 22 | 35,650 |
No | output | 1 | 17,825 | 22 | 35,651 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers a and m. Calculate the number of integers x such that 0 β€ x < m and \gcd(a, m) = \gcd(a + x, m).
Note: \gcd(a, b) is the greatest common divisor of a and b.
Input
The first line contains the single integer T (1 β€ T β€ 50) β the number of test cases.
Next T lines contain test cases β one per line. Each line contains two integers a and m (1 β€ a < m β€ 10^{10}).
Output
Print T integers β one per test case. For each test case print the number of appropriate x-s.
Example
Input
3
4 9
5 10
42 9999999967
Output
6
1
9999999966
Note
In the first test case appropriate x-s are [0, 1, 3, 4, 6, 7].
In the second test case the only appropriate x is 0.
Submitted Solution:
```
def gcd(x, y):
if x > y:
small = y
else:
small = x
for i in range(1, small+1):
if((x % i == 0) and (y % i == 0)):
gc = i
return gc
n=int(input())
res=[]
for i in range(n):
count=0
a,m=[int(x) for x in input().split()]
if m%2!=0 and m%3!=0 and m%5!=0 and m%7!=0:
count=m-1
else:
agm=gcd(a,m)
for j in range(m):
if gcd(a+j,m)==agm:
count+=1
res.append(count)
for i in res:
print(i)
``` | instruction | 0 | 17,826 | 22 | 35,652 |
No | output | 1 | 17,826 | 22 | 35,653 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,853 | 22 | 35,706 |
Tags: brute force, math, number theory
Correct Solution:
```
import sys
readline = sys.stdin.readline
from collections import Counter
def gcd(a, b):
while b: a, b = b, a % b
return a
def isPrimeMR(n):
d = n - 1
d = d // (d & -d)
L = [2]
for a in L:
t = d
y = pow(a, t, n)
if y == 1: continue
while y != n - 1:
y = (y * y) % n
if y == 1 or t == n - 1: return 0
t <<= 1
return 1
def findFactorRho(n):
m = 1 << n.bit_length() // 8
for c in range(1, 99):
f = lambda x: (x * x + c) % n
y, r, q, g = 2, 1, 1, 1
while g == 1:
x = y
for i in range(r):
y = f(y)
k = 0
while k < r and g == 1:
ys = y
for i in range(min(m, r - k)):
y = f(y)
q = q * abs(x - y) % n
g = gcd(q, n)
k += m
r <<= 1
if g == n:
g = 1
while g == 1:
ys = f(ys)
g = gcd(abs(x - ys), n)
if g < n:
if isPrimeMR(g): return g
elif isPrimeMR(n // g): return n // g
return findFactorRho(g)
def primeFactor(n):
i = 2
ret = {}
rhoFlg = 0
while i*i <= n:
k = 0
while n % i == 0:
n //= i
k += 1
if k: ret[i] = k
i += 1 + i % 2
if i == 101 and n >= 2 ** 20:
while n > 1:
if isPrimeMR(n):
ret[n], n = 1, 1
else:
rhoFlg = 1
j = findFactorRho(n)
k = 0
while n % j == 0:
n //= j
k += 1
ret[j] = k
if n > 1: ret[n] = 1
if rhoFlg: ret = {x: ret[x] for x in sorted(ret)}
return ret
T = int(readline())
Ans = [None]*T
for qu in range(T):
p, q = map(int, readline().split())
if p % q:
Ans[qu] = p
else:
k = p//q
cp = Counter(primeFactor(k))
cq = Counter(primeFactor(q))
for k, v in cq.items():
cp[k] += v
res = 1
for k, v in cp.items():
if cq[k] == 0:
continue
can = p//pow(k, cp[k] - cq[k] + 1)
if can % q:
res = max(res, can)
Ans[qu] = res
print('\n'.join(map(str, Ans)))
``` | output | 1 | 17,853 | 22 | 35,707 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,854 | 22 | 35,708 |
Tags: brute force, math, number theory
Correct Solution:
```
from math import pow
def pizda(n):
i = 2
primfac = []
while i * i <= n:
while n % i == 0:
primfac.append(i)
n = n // i
i = i + 1
if n > 1:
primfac.append(n)
return primfac
t=int(input())
for j in range(t):
p,q = map(int,input().split())
if p%q!=0:
print(p)
else:
a = pizda(q)
b = []
f = [a[0]]
for i in range(1, len(a)):
if a[i] == a[i - 1]:
f.append(a[i])
else:
b.append(f)
f = [a[i]]
b.append(f)
a = []
c = []
for i in range(len(b)):
a.append(b[i][0])
c.append(len(b[i]))
b = [0 for i in range(len(a))]
p1 = p
for i in range(len(a)):
while p % a[i] == 0:
p = p // a[i]
b[i] += 1
for i in range(len(a)):
b[i] = max(0, b[i] - c[i] + 1)
for i in range(len(a)):
b[i] = a[i]**b[i]
b.sort()
print(p1 // b[0])
``` | output | 1 | 17,854 | 22 | 35,709 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,855 | 22 | 35,710 |
Tags: brute force, math, number theory
Correct Solution:
```
for _ in range(int(input())):
p, q = map(int,input().split())
res= q
arr=[];j=2
while j*j <= res:
if res % j == 0:
arr.append(j)
while res % j == 0:
res = res // j
j += 1
if res > 1:
arr.append(res)
res = 1
for k in arr:
y = p
while y % q == 0:
y = y // k
res = max(res, y)
print(res)
``` | output | 1 | 17,855 | 22 | 35,711 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,856 | 22 | 35,712 |
Tags: brute force, math, number theory
Correct Solution:
```
def facts(n):
r=[n]
i=2
while(i*i<=n):
if(n%i==0):
r.append(i)
if(n//i!=i):
r.append(n//i)
i+=1
return r
for i in range(int(input())):
a,b=map(int,input().split())
if(a%b!=0):
print(a)
else:
c=1
x=facts(b)
#print(x)
for i in x:
n=a
while(n%i==0):
n//=i
if(n%b!=0):
c=max(c,n)
break
print(c)
``` | output | 1 | 17,856 | 22 | 35,713 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,857 | 22 | 35,714 |
Tags: brute force, math, number theory
Correct Solution:
```
for _ in range(int(input())):
a,b = list(map(int,input().split()))
if a%b != 0:
print(a)
continue
bDevVal = []
bDevAm = []
iter = 2
bc = b
while iter*iter <= bc:
if bc%iter == 0:
bDevVal.append(iter)
bDevAm.append(0)
while bc%iter == 0:
bDevAm[-1]+=1
bc//=iter
iter+=1
# print(bDevVal)
# print(bDevAm)
if bc > 1:
bDevVal.append(bc)
bDevAm.append(1)
m = 1
for i in range(len(bDevVal)):
ac = a
while ac%bDevVal[i] == 0:
ac//=bDevVal[i]
ac*=(bDevVal[i]**(bDevAm[i]-1))
m = max(ac,m)
print(m)
``` | output | 1 | 17,857 | 22 | 35,715 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,858 | 22 | 35,716 |
Tags: brute force, math, number theory
Correct Solution:
```
for _ in " "*int(input()):
p,q=map(int,input().split())
if p%q:
print(p)
else:
lst=[]
i=2
while i*i<=q:
if not q%i:
lst.append(i)
if i!=q//i:
lst.append(q//i)
i+=1
lst.append(q)
num=1
# print(lst)
for i in lst:
ele=p
# print(ele,i)
# print()
while not ele%i:
ele//=i
# print(ele)
if ele%q:
num=max(num,ele)
print(num)
``` | output | 1 | 17,858 | 22 | 35,717 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,859 | 22 | 35,718 |
Tags: brute force, math, number theory
Correct Solution:
```
from collections import defaultdict
primes = {i for i in range(2, 10**5)}
for i in range(2, 10**5):
if i in primes:
for j in range(2*i, 10**5, i):
if j in primes:
primes.remove(j)
primes = list(primes)
primes.sort()
def solve():
p, q = map(int, input().split())
P = p
if p % q != 0:
print(p)
return
#p%q == 0
pdiv = defaultdict(int)
qdiv = defaultdict(int)
for i in range(2, q+1):
if i*i > q:
break
if q % i == 0:
while q % i == 0:
pdiv[i] += 1
qdiv[i] += 1
p //= i
q //= i
while p % i == 0:
pdiv[i] += 1
p //= i
if q > 1:
qdiv[q] += 1
while p % q == 0:
pdiv[q] += 1
p //= q
ans = -1
for v in pdiv.keys():
ans = max(ans, P//(v**(pdiv[v] - qdiv[v] + 1)))
print(ans)
return
def main():
testcase = int(input())
for i in range(testcase):
solve()
return
if __name__ == "__main__":
main()
``` | output | 1 | 17,859 | 22 | 35,719 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. | instruction | 0 | 17,860 | 22 | 35,720 |
Tags: brute force, math, number theory
Correct Solution:
```
import sys
from math import gcd,sqrt,ceil,log2
from collections import defaultdict,Counter,deque
from bisect import bisect_left,bisect_right
import math
sys.setrecursionlimit(2*10**5+10)
import heapq
from itertools import permutations
# input=sys.stdin.readline
# def print(x):
# sys.stdout.write(str(x)+"\n")
# sys.stdin = open('input.txt', 'r')
# sys.stdout = open('output.txt', 'w')
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 8192
aa='abcdefghijklmnopqrstuvwxyz'
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
# import sys
# import io, os
# input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline
def get_sum(bit,i):
s = 0
i+=1
while i>0:
s+=bit[i]
i-=i&(-i)
return s
def update(bit,n,i,v):
i+=1
while i<=n:
bit[i]+=v
i+=i&(-i)
def modInverse(b,m):
g = math.gcd(b, m)
if (g != 1):
return -1
else:
return pow(b, m - 2, m)
def primeFactors(n):
sa = []
# sa.add(n)
while n % 2 == 0:
sa.append(2)
n = n // 2
for i in range(3,int(math.sqrt(n))+1,2):
while n % i== 0:
sa.append(i)
n = n // i
# sa.add(n)
if n > 2:
sa.append(n)
return sa
def seive(n):
pri = [True]*(n+1)
p = 2
while p*p<=n:
if pri[p] == True:
for i in range(p*p,n+1,p):
pri[i] = False
p+=1
return pri
def check_prim(n):
if n<0:
return False
for i in range(2,int(sqrt(n))+1):
if n%i == 0:
return False
return True
def getZarr(string, z):
n = len(string)
# [L,R] make a window which matches
# with prefix of s
l, r, k = 0, 0, 0
for i in range(1, n):
# if i>R nothing matches so we will calculate.
# Z[i] using naive way.
if i > r:
l, r = i, i
# R-L = 0 in starting, so it will start
# checking from 0'th index. For example,
# for "ababab" and i = 1, the value of R
# remains 0 and Z[i] becomes 0. For string
# "aaaaaa" and i = 1, Z[i] and R become 5
while r < n and string[r - l] == string[r]:
r += 1
z[i] = r - l
r -= 1
else:
# k = i-L so k corresponds to number which
# matches in [L,R] interval.
k = i - l
# if Z[k] is less than remaining interval
# then Z[i] will be equal to Z[k].
# For example, str = "ababab", i = 3, R = 5
# and L = 2
if z[k] < r - i + 1:
z[i] = z[k]
# For example str = "aaaaaa" and i = 2,
# R is 5, L is 0
else:
# else start from R and check manually
l = i
while r < n and string[r - l] == string[r]:
r += 1
z[i] = r - l
r -= 1
def search(text, pattern):
# Create concatenated string "P$T"
concat = pattern + "$" + text
l = len(concat)
z = [0] * l
getZarr(concat, z)
ha = []
for i in range(l):
if z[i] == len(pattern):
ha.append(i - len(pattern) - 1)
return ha
# n,k = map(int,input().split())
# l = list(map(int,input().split()))
#
# n = int(input())
# l = list(map(int,input().split()))
#
# hash = defaultdict(list)
# la = []
#
# for i in range(n):
# la.append([l[i],i+1])
#
# la.sort(key = lambda x: (x[0],-x[1]))
# ans = []
# r = n
# flag = 0
# lo = []
# ha = [i for i in range(n,0,-1)]
# yo = []
# for a,b in la:
#
# if a == 1:
# ans.append([r,b])
# # hash[(1,1)].append([b,r])
# lo.append((r,b))
# ha.pop(0)
# yo.append([r,b])
# r-=1
#
# elif a == 2:
# # print(yo,lo)
# # print(hash[1,1])
# if lo == []:
# flag = 1
# break
# c,d = lo.pop(0)
# yo.pop(0)
# if b>=d:
# flag = 1
# break
# ans.append([c,b])
# yo.append([c,b])
#
#
#
# elif a == 3:
#
# if yo == []:
# flag = 1
# break
# c,d = yo.pop(0)
# if b>=d:
# flag = 1
# break
# if ha == []:
# flag = 1
# break
#
# ka = ha.pop(0)
#
# ans.append([ka,b])
# ans.append([ka,d])
# yo.append([ka,b])
#
# if flag:
# print(-1)
# else:
# print(len(ans))
# for a,b in ans:
# print(a,b)
def mergeIntervals(arr):
# Sorting based on the increasing order
# of the start intervals
arr.sort(key = lambda x: x[0])
# array to hold the merged intervals
m = []
s = -10000
max = -100000
for i in range(len(arr)):
a = arr[i]
if a[0] > max:
if i != 0:
m.append([s,max])
max = a[1]
s = a[0]
else:
if a[1] >= max:
max = a[1]
#'max' value gives the last point of
# that particular interval
# 's' gives the starting point of that interval
# 'm' array contains the list of all merged intervals
if max != -100000 and [s, max] not in m:
m.append([s, max])
return m
class SortedList:
def __init__(self, iterable=[], _load=200):
"""Initialize sorted list instance."""
values = sorted(iterable)
self._len = _len = len(values)
self._load = _load
self._lists = _lists = [values[i:i + _load] for i in range(0, _len, _load)]
self._list_lens = [len(_list) for _list in _lists]
self._mins = [_list[0] for _list in _lists]
self._fen_tree = []
self._rebuild = True
def _fen_build(self):
"""Build a fenwick tree instance."""
self._fen_tree[:] = self._list_lens
_fen_tree = self._fen_tree
for i in range(len(_fen_tree)):
if i | i + 1 < len(_fen_tree):
_fen_tree[i | i + 1] += _fen_tree[i]
self._rebuild = False
def _fen_update(self, index, value):
"""Update `fen_tree[index] += value`."""
if not self._rebuild:
_fen_tree = self._fen_tree
while index < len(_fen_tree):
_fen_tree[index] += value
index |= index + 1
def _fen_query(self, end):
"""Return `sum(_fen_tree[:end])`."""
if self._rebuild:
self._fen_build()
_fen_tree = self._fen_tree
x = 0
while end:
x += _fen_tree[end - 1]
end &= end - 1
return x
def _fen_findkth(self, k):
"""Return a pair of (the largest `idx` such that `sum(_fen_tree[:idx]) <= k`, `k - sum(_fen_tree[:idx])`)."""
_list_lens = self._list_lens
if k < _list_lens[0]:
return 0, k
if k >= self._len - _list_lens[-1]:
return len(_list_lens) - 1, k + _list_lens[-1] - self._len
if self._rebuild:
self._fen_build()
_fen_tree = self._fen_tree
idx = -1
for d in reversed(range(len(_fen_tree).bit_length())):
right_idx = idx + (1 << d)
if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
idx = right_idx
k -= _fen_tree[idx]
return idx + 1, k
def _delete(self, pos, idx):
"""Delete value at the given `(pos, idx)`."""
_lists = self._lists
_mins = self._mins
_list_lens = self._list_lens
self._len -= 1
self._fen_update(pos, -1)
del _lists[pos][idx]
_list_lens[pos] -= 1
if _list_lens[pos]:
_mins[pos] = _lists[pos][0]
else:
del _lists[pos]
del _list_lens[pos]
del _mins[pos]
self._rebuild = True
def _loc_left(self, value):
"""Return an index pair that corresponds to the first position of `value` in the sorted list."""
if not self._len:
return 0, 0
_lists = self._lists
_mins = self._mins
lo, pos = -1, len(_lists) - 1
while lo + 1 < pos:
mi = (lo + pos) >> 1
if value <= _mins[mi]:
pos = mi
else:
lo = mi
if pos and value <= _lists[pos - 1][-1]:
pos -= 1
_list = _lists[pos]
lo, idx = -1, len(_list)
while lo + 1 < idx:
mi = (lo + idx) >> 1
if value <= _list[mi]:
idx = mi
else:
lo = mi
return pos, idx
def _loc_right(self, value):
"""Return an index pair that corresponds to the last position of `value` in the sorted list."""
if not self._len:
return 0, 0
_lists = self._lists
_mins = self._mins
pos, hi = 0, len(_lists)
while pos + 1 < hi:
mi = (pos + hi) >> 1
if value < _mins[mi]:
hi = mi
else:
pos = mi
_list = _lists[pos]
lo, idx = -1, len(_list)
while lo + 1 < idx:
mi = (lo + idx) >> 1
if value < _list[mi]:
idx = mi
else:
lo = mi
return pos, idx
def add(self, value):
"""Add `value` to sorted list."""
_load = self._load
_lists = self._lists
_mins = self._mins
_list_lens = self._list_lens
self._len += 1
if _lists:
pos, idx = self._loc_right(value)
self._fen_update(pos, 1)
_list = _lists[pos]
_list.insert(idx, value)
_list_lens[pos] += 1
_mins[pos] = _list[0]
if _load + _load < len(_list):
_lists.insert(pos + 1, _list[_load:])
_list_lens.insert(pos + 1, len(_list) - _load)
_mins.insert(pos + 1, _list[_load])
_list_lens[pos] = _load
del _list[_load:]
self._rebuild = True
else:
_lists.append([value])
_mins.append(value)
_list_lens.append(1)
self._rebuild = True
def discard(self, value):
"""Remove `value` from sorted list if it is a member."""
_lists = self._lists
if _lists:
pos, idx = self._loc_right(value)
if idx and _lists[pos][idx - 1] == value:
self._delete(pos, idx - 1)
def remove(self, value):
"""Remove `value` from sorted list; `value` must be a member."""
_len = self._len
self.discard(value)
if _len == self._len:
raise ValueError('{0!r} not in list'.format(value))
def pop(self, index=-1):
"""Remove and return value at `index` in sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
value = self._lists[pos][idx]
self._delete(pos, idx)
return value
def bisect_left(self, value):
"""Return the first index to insert `value` in the sorted list."""
pos, idx = self._loc_left(value)
return self._fen_query(pos) + idx
def bisect_right(self, value):
"""Return the last index to insert `value` in the sorted list."""
pos, idx = self._loc_right(value)
return self._fen_query(pos) + idx
def count(self, value):
"""Return number of occurrences of `value` in the sorted list."""
return self.bisect_right(value) - self.bisect_left(value)
def __len__(self):
"""Return the size of the sorted list."""
return self._len
def __getitem__(self, index):
"""Lookup value at `index` in sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
return self._lists[pos][idx]
def __delitem__(self, index):
"""Remove value at `index` from sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
self._delete(pos, idx)
def __contains__(self, value):
"""Return true if `value` is an element of the sorted list."""
_lists = self._lists
if _lists:
pos, idx = self._loc_left(value)
return idx < len(_lists[pos]) and _lists[pos][idx] == value
return False
def __iter__(self):
"""Return an iterator over the sorted list."""
return (value for _list in self._lists for value in _list)
def __reversed__(self):
"""Return a reverse iterator over the sorted list."""
return (value for _list in reversed(self._lists) for value in reversed(_list))
def __repr__(self):
"""Return string representation of sorted list."""
return 'SortedList({0})'.format(list(self))
def ncr(n, r, p):
num = den = 1
for i in range(r):
num = (num * (n - i)) % p
den = (den * (i + 1)) % p
return (num * pow(den,
p - 2, p)) % p
def sol(n):
seti = set()
for i in range(1,int(sqrt(n))+1):
if n%i == 0:
seti.add(n//i)
seti.add(i)
return seti
def lcm(a,b):
return (a*b)//gcd(a,b)
#
# n,p = map(int,input().split())
#
# s = input()
#
# if n <=2:
# if n == 1:
# pass
# if n == 2:
# pass
# i = n-1
# idx = -1
# while i>=0:
# z = ord(s[i])-96
# k = chr(z+1+96)
# flag = 1
# if i-1>=0:
# if s[i-1]!=k:
# flag+=1
# else:
# flag+=1
# if i-2>=0:
# if s[i-2]!=k:
# flag+=1
# else:
# flag+=1
# if flag == 2:
# idx = i
# s[i] = k
# break
# if idx == -1:
# print('NO')
# exit()
# for i in range(idx+1,n):
# if
#
t = int(input())
for _ in range(t):
p,q = map(int,input().split())
# pr1 = primeFactors(p)
pr1 = primeFactors(q)
hash = defaultdict(int)
if p%q!=0:
print(p)
continue
maxi = 1
for i in pr1:
hash[i]+=1
for i in hash:
cnt = 0
temp = p
while temp%i == 0:
temp//=i
maxi = max(temp*(i**(hash[i]-1)),maxi)
print(maxi)
``` | output | 1 | 17,860 | 22 | 35,721 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
primes = []
max_ = int(1e5)
prime = [True] * max_
prime[0] = prime[1] = False
i = 2
while i * i <= max_:
if prime[i]:
for j in range(2 * i, max_, i):
prime[j] = False
i += 1
for i in range(max_):
if prime[i]:
primes.append(i)
t = int(input())
while t > 0:
p, q = map(int, input().split())
xx = q
to_ = []
for i in primes:
if xx % i == 0:
to_.append(i)
while xx % i == 0:
xx //= i
if xx == 1:
break
if xx != 1:
to_.append(xx)
if p % q != 0:
print(p)
else:
ans = 1
for j in to_:
p1 = p
while p1 % q == 0:
p1 //= j
if p1 > ans:
ans = p1
print(ans)
t -= 1
``` | instruction | 0 | 17,861 | 22 | 35,722 |
Yes | output | 1 | 17,861 | 22 | 35,723 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
import sys
from sys import stdin
tt = int(stdin.readline())
for loop in range(tt):
p,q = map(int,stdin.readline().split())
if p % q != 0:
print (p)
continue
bk = {}
for i in range(2,10**5):
if q % i == 0:
bk[i] = 0
while q % i == 0:
bk[i] += 1
q //= i
if bk == 1:
break
if q != 1:
bk[q] = 1
#print (bk)
ans = float("-inf")
for i in bk:
cnt = 0
tp = p
while tp % i == 0:
tp //= i
cnt += 1
ans = max(ans , p // (i**(cnt-bk[i]+1)))
print (ans)
``` | instruction | 0 | 17,862 | 22 | 35,724 |
Yes | output | 1 | 17,862 | 22 | 35,725 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
t = int(input())
for _ in range(t):
p, q = map(int, input().split())
out = []
nq = q
for test in range(2, 10**5):
if q % test == 0:
p2 = p
while p2 % q == 0:
p2 //= test
out.append(p2)
while nq % test == 0:
nq //= test
if nq > 1:
while p % q == 0:
p //= nq
out.append(p)
print(max(out))
``` | instruction | 0 | 17,863 | 22 | 35,726 |
Yes | output | 1 | 17,863 | 22 | 35,727 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
def divisors(q):
d = 2
while(d<=int(q**0.5)+1):
if q%d == 0:
div.append(d)
if q//d != d:
div.append(q//d)
d+=1
div.append(q)
if __name__ == '__main__':
for _ in range(int(input())):
p,q = map(int,input().split())
if p%q !=0:
print(p)
else:
div = []
for i in range(2,int(q**0.5)+1):
if q%i==0:
div.append(i)
div.append(q//i)
div.append(q)
ans = 1
for i in div:
temp = p
while temp%i==0:
temp = temp//i
if temp%q !=0:
ans = max(ans,temp)
break
print(ans)
``` | instruction | 0 | 17,864 | 22 | 35,728 |
Yes | output | 1 | 17,864 | 22 | 35,729 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
from math import sqrt
t = int(input())
for _ in range(t):
p, q = map(int, input().split())
if p % q:
print(p)
else:
i, fact, res = 2, {}, 1
if q % 2 == 0:
fact[2], prod = 0, 1
while q % 2 == 0:
fact[2] += 1
q /= 2
prod *= 2
prod /= 2
val = p
while val % 2 == 0: val /= 2
res = max(res, val * prod)
for i in range(3, int(sqrt(q)), 2):
if q % i == 0:
fact[i], prod = 0, 1
while q % i == 0:
fact[i] += 1
q /= i
prod *= i
prod /= i
val = p
while val % i == 0: val /= i
res = max(res, val * prod)
print(int(res))
``` | instruction | 0 | 17,865 | 22 | 35,730 |
No | output | 1 | 17,865 | 22 | 35,731 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
from math import gcd
def getdivs(n):
#[print(x, some(x, p)) for x in range(2, int(n ** 0.5) + 1) if not n % x]
divs = [x for x in range(2, int(n ** 0.5) + 1) if not n % x]
#for x in divs:
# divs.append(n // x)
ans = []
for x in divs:
ans.append(x ** some(x, p))
if ans == []:
#print('>>>>>', p, q, n, '<<<<<', ans, divs)
ans = [n]
divs = [n ** some(n, p)]
return [divs, ans]
def some(x, n):
ans = 0
y = x
while n % x == 0:
x *= y
ans += 1
return ans
for _ in range(int(input())):
p, q = map(int, input().split())
if p % q == 0:
divs, dance = getdivs(gcd(p, q))
#print(']', divs, dance)
ans = p // divs[dance.index(min(dance))]
#ans = p // min(getdivs(gcd(p, q)))
else:
ans = p
print(ans)
``` | instruction | 0 | 17,866 | 22 | 35,732 |
No | output | 1 | 17,866 | 22 | 35,733 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
import math
t = int(input())
cp = []
for i in range(0, t):
p, q = map(int, input().split())
q0 = math.sqrt(q)
q0 = math.ceil(q0) #ΠΎΠΊΡΡΠ³Π»ΡΠ΅ΠΌ
if q > p or p % q != 0:
print(p)
continue
for d in range(1, q0 + 1): #ΡΠΈΠΊΠ» Ρ Π΄ΠΎΠ±Π°Π²Π»Π΅Π½ΠΈΠ΅ΠΌ ΠΏΡΠΎΡΡΡΡ
Π΄Π΅Π»ΠΈΡΠ΅Π»Π΅ΠΉ q
if q % d == 0:
qd = q // d
if qd != d:
cp.append(d)
cp.append(qd)
else:
cp.append(d)
cp.sort()
cp.reverse()
# print(cp)
x0 = 1
p0 = p
for j in cp:
while p0 % j == 0:
p0 = p0 // j
if p0 % j != 0:
x0 = max(p0, x0)
continue
else:
break
cp.clear()
print(x0)
``` | instruction | 0 | 17,867 | 22 | 35,734 |
No | output | 1 | 17,867 | 22 | 35,735 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.
To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that:
* p_i is divisible by x_i;
* x_i is not divisible by q_i.
Oleg is really good at division and managed to find all the answers quickly, how about you?
Input
The first line contains an integer t (1 β€ t β€ 50) β the number of pairs.
Each of the following t lines contains two integers p_i and q_i (1 β€ p_i β€ 10^{18}; 2 β€ q_i β€ 10^{9}) β the i-th pair of integers.
Output
Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i.
One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints.
Example
Input
3
10 4
12 6
179 822
Output
10
4
179
Note
For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4.
For the second pair, where p_2 = 12 and q_2 = 6, note that
* 12 is not a valid x_2, since 12 is divisible by q_2 = 6;
* 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6.
The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6.
Submitted Solution:
```
def solution(p, q):
max_num = -1
if p % q != 0:
return p
x = q
while x > 1:
for div in range(2,q+1):
if x % div == 0:
x = x // div
if p % x == 0:
return x
break
return -1
t = int(input())
for tt in range(t):
p, q = map(int, input().split())
print(solution(p, q))
``` | instruction | 0 | 17,868 | 22 | 35,736 |
No | output | 1 | 17,868 | 22 | 35,737 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,011 | 22 | 36,022 |
Tags: math, number theory
Correct Solution:
```
n,x=map(int,input().split())
l=list(map(int,input().split()))
from collections import Counter
l.sort()
gcd=sum(l[:n-1])
m=10**9+7
s=sum(l)
l1=[]
for i in l:
l1.append(s-i-gcd)
c=Counter(l1)
l2=[]
for i in c:
l2.append(i)
l2.sort()
z=True
while z:
z=False
for i in c.copy():
a=c[i]
b=0
while a%x==0 and a!=0:
z=True
a//=x
b+=1
if z:
c[i+b]+=a
c[i]=0
y=l[-1]
for i in c:
if c[i]:
y=min(y,i)
gcd+=y
print(pow(x,min(s,gcd),m))
``` | output | 1 | 18,011 | 22 | 36,023 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,012 | 22 | 36,024 |
Tags: math, number theory
Correct Solution:
```
import math
pri=pow(10,9)+7
n,x=map(int,input().split())
z=list(map(int,input().split()))
sumr=sum(z)
gcd=sumr-max(z)
t=max(z)
ans=z.copy()
for i in range(len(ans)):
ans[i]=t-ans[i]
ans.sort()
temp=[]
count=1
from heapq import *
temp=[]
for i in range(1,len(ans)):
if(ans[i]==ans[i-1]):
count+=1
else:
temp.append([ans[i-1],count])
count=1
temp.append([ans[-1],count])
pq=[]
from collections import *
al=defaultdict(int)
for i in range(len(temp)):
al[temp[i][0]]=temp[i][1]
heappush(pq,temp[i][0])
while(len(pq)):
se=heappop(pq)
r=al[se]
c1=0
while(r%x==0 and r>0):
r=r//x
c1+=1
if(c1==0):
ty=min(se+gcd,sumr)
print(pow(x,ty,pri))
exit()
else:
if(al[se+c1]):
al[se+c1]+=r
else:
heappush(pq,se+c1)
al[se+c1]=r
``` | output | 1 | 18,012 | 22 | 36,025 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,013 | 22 | 36,026 |
Tags: math, number theory
Correct Solution:
```
n, x = map(int, input().split())
t = list(map(int, input().split()))
b = max(t)
k = t.count(b)
r = sum(t)
s = r - b
while k % x == 0:
b -= 1
s += 1
k = k // x + t.count(b)
print(pow(x, min(r, s), 1000000007))
# Made By Mostafa_Khaled
``` | output | 1 | 18,013 | 22 | 36,027 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,014 | 22 | 36,028 |
Tags: math, number theory
Correct Solution:
```
def Solve(x, a_i):
max_a_i = max(a_i)
count = a_i.count(max_a_i)
A = sum(a_i)
v = A - max_a_i
while count % x == 0:
max_a_i -= 1
v +=1
count = count//x + a_i.count(max_a_i)
return pow(x, min(v, A), 1000000007)
n, x = map(int, input().split())
a_i = list(map(int, input().split()))
print(Solve(x, a_i))
``` | output | 1 | 18,014 | 22 | 36,029 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,015 | 22 | 36,030 |
Tags: math, number theory
Correct Solution:
```
modulus = 10 ** 9 + 7
def main():
n, x = map(int, input().split())
arr = list(map(int, input().split()))
total = sum(arr)
powers = [total - x for x in arr]
powers.sort(reverse=True)
while True:
low = powers[len(powers) - 1]
cnt = 0
while len(powers) > 0 and powers[len(powers) - 1] == low:
cnt += 1
powers.pop()
if cnt % x == 0:
cnt = cnt // x
for i in range(cnt):
powers.append(low + 1)
else:
low = min(low, total)
print(pow(x, low, modulus))
exit(0)
if __name__ == "__main__":
main()
``` | output | 1 | 18,015 | 22 | 36,031 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,016 | 22 | 36,032 |
Tags: math, number theory
Correct Solution:
```
MOD = 10**9+7
def pow(x, n, MOD):
if(n==0):
return 1
if(n%2==0):
a = pow(x, n//2, MOD)
return a*a % MOD
else:
a = pow(x, n//2, MOD)
return (x*a*a) % MOD
n, x = [int(x) for x in input().split()]
a = [int(x) for x in input().split()]
s = sum(a)
d = []
for el in a:
d.append(s-el)
d.sort()
i=0
gcd = d[0]
c=0
while i<n and d[i]==gcd:
c+=1
i+=1
while c%x==0:
c=c//x
gcd+=1
while i<n and d[i]==gcd:
c+=1
i+=1
if gcd>s:
gcd = s
print( pow(x, gcd, MOD) )
``` | output | 1 | 18,016 | 22 | 36,033 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,017 | 22 | 36,034 |
Tags: math, number theory
Correct Solution:
```
n, x = map(int, input().split())
t = list(map(int, input().split()))
b = max(t)
k = t.count(b)
r = sum(t)
s = r - b
while k % x == 0:
b -= 1
s += 1
k = k // x + t.count(b)
print(pow(x, min(r, s), 1000000007))
``` | output | 1 | 18,017 | 22 | 36,035 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,018 | 22 | 36,036 |
Tags: math, number theory
Correct Solution:
```
#!/usr/bin/python3
def readln(): return tuple(map(int, input().split()))
n, x = readln()
a = readln()
m = max(a)
ans = sum(a) - m
cnt = {}
for v in a:
cnt[m - v] = cnt.get(m - v, 0) + 1
while True:
k = min(cnt.keys())
v = cnt[k]
c = 0
while v % x == 0:
c += 1
v //= x
#print(cnt, k, cnt[k], c, v)
if c:
cnt[k + c] = cnt.get(k + c, 0) + v
cnt.pop(k)
else:
break
#print(cnt)
print(pow(x, ans + min(min(cnt.keys()), m), 10**9 + 7))
``` | output | 1 | 18,018 | 22 | 36,037 |
Provide tags and a correct Python 2 solution for this coding contest problem.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1. | instruction | 0 | 18,019 | 22 | 36,038 |
Tags: 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 in_num():
return int(raw_input())
def in_arr():
return map(int,raw_input().split())
def pr_num(n):
stdout.write(str(n)+'\n')
def pr_arr(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+
n,x=in_arr()
l=in_arr()
sm=sum(l)
v=l[-1]
c=0
while v:
while l and v==l[-1]:
c+=1
l.pop()
if c%x:
break
c/=x
v-=1
pr_num(pow(x,sm-v,mod))
``` | output | 1 | 18,019 | 22 | 36,039 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
def Solve(x, a_i):
v = max(a_i)
count = a_i.count(v)
A = sum(a_i)
s = A - v
while count % x == 0:
s +=1
v -= 1
count += count//x + a_i.count(v)
return pow(x, min(s, A), 1000000007)
n, x = map(int, input().split())
a_i = list(map(int, input().split()))
print(Solve(x, a_i))
``` | instruction | 0 | 18,020 | 22 | 36,040 |
Yes | output | 1 | 18,020 | 22 | 36,041 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
input_str = input()
n, x = int(input_str.split()[0]), int(input_str.split()[1])
input_str = input()
a = input_str.split()
s = 0
for i in range(n):
a[i] = int(a[i])
#a.append(1000000000)
s += a[i]
'''
a = []
for i in range(n):
if i<65536:
temp = 1
else:
temp = 0#random.randint(0, 1000000000)
s += temp
a.append(temp)#(557523474, 999999999))'''
sum_a = [s-a[i] for i in range(n)]
minimum = min(sum_a)
res = pow(x, minimum, 1000000007)#FastPow(x, minimum)#x**minimum
sum_xa = 0
s_new = s - minimum
for i in range(n):
sum_xa += pow(x, s_new - a[i], 1000000007)#FastPow(x, s_new - a[i])
deg = 0
deg_zn = s-minimum
ts = sum_xa
while sum_xa%1==0 and deg_zn:
sum_xa /= x
deg += 1
deg_zn -= 1
# print (deg, sum_xa%1==0)
#if (n, x) == (98304, 2):
# print (minimum, s, ts, deg)
if deg:
res *= pow(x, deg-1 if sum_xa%1!=0 else deg)
print (res%1000000007)
``` | instruction | 0 | 18,021 | 22 | 36,042 |
Yes | output | 1 | 18,021 | 22 | 36,043 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
n, x = map(int, input().split(' '))
arr = list(map(int, input().split(' ')))
sx = sum(arr)
adds = [sx - i for i in arr]
adds.sort()
while adds.count(adds[0]) % x == 0:
ct = adds.count(adds[0])
addsok = ct // x
adds = [adds[0]+1] * addsok + adds[ct:]
print(pow(x, min(min(adds), sx), (10**9+7)))
``` | instruction | 0 | 18,022 | 22 | 36,044 |
Yes | output | 1 | 18,022 | 22 | 36,045 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
n, x = map(int, input().split())
a = list(map(int, input().split()))
total = sum(a)
biggest = max(a)
exp = total - biggest
hm = a.count(biggest)
while hm%x == 0:
biggest -= 1
exp += 1
hm = hm//x + a.count(biggest)
if exp > total:
exp = total
print(pow(x, exp, 1000000007))
``` | instruction | 0 | 18,023 | 22 | 36,046 |
Yes | output | 1 | 18,023 | 22 | 36,047 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
n, x = map(int, input().split())
t = list(map(int, input().split()))
b = max(t)
k = t.count(b)
s = sum(t) - b
while k % x == 0:
b -= 1
s += 1
k = k // x + t.count(b)
print(pow(x, s, 1000000007))
``` | instruction | 0 | 18,024 | 22 | 36,048 |
No | output | 1 | 18,024 | 22 | 36,049 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
input_str = input()
n, x = int(input_str.split()[0]), int(input_str.split()[1])
input_str = input()
a = input_str.split()
s = 0
for i in range(n):
a[i] = int(a[i])
#a.append(1000000000)
s += a[i]
'''a = []
for i in range(n):
temp = 0#random.randint(0, 1000000000)
s += temp
a.append(temp)#(557523474, 999999999))'''
sum_a = [s-a[i] for i in range(n)]
minimum = min(sum_a)
res = pow(x, minimum, 1000000007)#FastPow(x, minimum)#x**minimum
sum_xa = 0
s_new = s - minimum
for i in range(n):
sum_xa += pow(x, s_new - a[i], 1000000007)#FastPow(x, s_new - a[i])
deg = 0
while not sum_xa%1:
sum_xa /= x
deg += 1
if (n, x) == (98304, 2):
print (deg, pow(x, s-minimum), s, minimum)
if deg:
res *= pow(x, deg-1)
print (res%1000000007)
``` | instruction | 0 | 18,025 | 22 | 36,050 |
No | output | 1 | 18,025 | 22 | 36,051 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
#!/usr/bin/python3
def readln(): return tuple(map(int, input().split()))
n, x = readln()
a = readln()
m = max(a)
ans = sum(a) - m
cnt = {}
for v in a:
cnt[m - v] = cnt.get(m - v, 0) + 1
while True:
k = min(cnt.keys())
v = cnt[k]
c = 0
while v and v % x == 0:
c += 1
v //= 2
if c:
ans += c
cnt[k + x * c] = cnt.get(k + x * c, 0) + c
if v == 0:
cnt.pop(k)
else:
cnt[k] = v
else:
break
print(pow(x, ans, 10**9 + 7))
``` | instruction | 0 | 18,026 | 22 | 36,052 |
No | output | 1 | 18,026 | 22 | 36,053 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Simon has a prime number x and an array of non-negative integers a1, a2, ..., an.
Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction.
Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7).
Input
The first line contains two positive integers n and x (1 β€ n β€ 105, 2 β€ x β€ 109) β the size of the array and the prime number.
The second line contains n space-separated integers a1, a2, ..., an (0 β€ a1 β€ a2 β€ ... β€ an β€ 109).
Output
Print a single number β the answer to the problem modulo 1000000007 (109 + 7).
Examples
Input
2 2
2 2
Output
8
Input
3 3
1 2 3
Output
27
Input
2 2
29 29
Output
73741817
Input
4 5
0 0 0 0
Output
1
Note
In the first sample <image>. Thus, the answer to the problem is 8.
In the second sample, <image>. The answer to the problem is 27, as 351 = 13Β·27, 729 = 27Β·27.
In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817.
In the fourth sample <image>. Thus, the answer to the problem is 1.
Submitted Solution:
```
import math as m
n,x=map(int,input().split())
arr=[int(i) for i in input().split()]
s=sum(arr)
l=s-max(arr)
powers=[]
cnt=0
for i in arr:
if s-i == l:
cnt+=1
l=l+cnt//x
mod=10**9+7
print(pow(x,l,mod))
``` | instruction | 0 | 18,027 | 22 | 36,054 |
No | output | 1 | 18,027 | 22 | 36,055 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Arthur and Alexander are number busters. Today they've got a competition.
Arthur took a group of four integers a, b, w, x (0 β€ b < w, 0 < x < w) and Alexander took integer Ρ. Arthur and Alexander use distinct approaches to number bustings. Alexander is just a regular guy. Each second, he subtracts one from his number. In other words, he performs the assignment: c = c - 1. Arthur is a sophisticated guy. Each second Arthur performs a complex operation, described as follows: if b β₯ x, perform the assignment b = b - x, if b < x, then perform two consecutive assignments a = a - 1; b = w - (x - b).
You've got numbers a, b, w, x, c. Determine when Alexander gets ahead of Arthur if both guys start performing the operations at the same time. Assume that Alexander got ahead of Arthur if c β€ a.
Input
The first line contains integers a, b, w, x, c (1 β€ a β€ 2Β·109, 1 β€ w β€ 1000, 0 β€ b < w, 0 < x < w, 1 β€ c β€ 2Β·109).
Output
Print a single integer β the minimum time in seconds Alexander needs to get ahead of Arthur. You can prove that the described situation always occurs within the problem's limits.
Examples
Input
4 2 3 1 6
Output
2
Input
4 2 3 1 7
Output
4
Input
1 2 3 2 6
Output
13
Input
1 1 2 1 1
Output
0 | instruction | 0 | 18,033 | 22 | 36,066 |
Tags: binary search, math
Correct Solution:
```
import fileinput;
str = input();
sstr = str.split(' ')
a,b,w,x,c = int(sstr[0]),int(sstr[1]),int(sstr[2]),int(sstr[3]),int(sstr[4])
dif = c-a
up = max(0,int((dif*x-b)/(w-x)))
b = b+up*(w-x)
count = up
while dif > 0:
if (b >= x):
factor = min(int((b - x)/x)+1,dif)
dif = dif-factor
b = b - x*factor
else:
factor = int( (x-b-1)/(w-x) )+1
b = factor*(w-x)+b
count = count+factor
print(count+(c-a))
``` | output | 1 | 18,033 | 22 | 36,067 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,224 | 22 | 36,448 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
fun = lambda num: -1 if num in [1, 2, 3, 5, 7, 11] else num//4 - num%2
[print(fun(int(input()))) for i in range(int(input()))]
# Made By Mostafa_Khaled
``` | output | 1 | 18,224 | 22 | 36,449 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,225 | 22 | 36,450 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
t = int(input())
for i in range(t):
n = int(input())
if n in [1, 2, 3, 5, 7, 11]:
print(-1)
else:
print((n // 4) - (n % 2))
``` | output | 1 | 18,225 | 22 | 36,451 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,226 | 22 | 36,452 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
import sys
readline = sys.stdin.readline
Q = int(readline())
Ans = [None]*Q
ans = [None, -1, -1, -1, 1, -1, 1, -1, 2, 1, 2, -1, 3, 2, 3, 2]
for qu in range(Q):
N = int(readline())
if N <= 15:
Ans[qu] = ans[N]
continue
if N%4 == 0:
Ans[qu] = N//4
elif N%4 == 1:
Ans[qu] = N//4 - 1
elif N%4 == 2:
Ans[qu] = N//4
else:
Ans[qu] = N//4 - 1
print('\n'.join(map(str, Ans)))
``` | output | 1 | 18,226 | 22 | 36,453 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,227 | 22 | 36,454 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
t = int(input())
for i in range(t):
n = int(input())
if n in [1, 2, 3, 5, 7, 11]:
print(-1)
elif n % 4 == 0:
print(n // 4)
elif n % 4 == 1:
print((n - 9) // 4 + 1)
elif n % 4 == 2:
print((n - 6) // 4 + 1)
elif n % 4 == 3:
print((n - 15) // 4 + 2)
``` | output | 1 | 18,227 | 22 | 36,455 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,228 | 22 | 36,456 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
for i in range(int(input())):
num = int(input())
print(-1 if num in [1, 2, 3, 5, 7, 11] else (num >> 2) - (num & 1))
``` | output | 1 | 18,228 | 22 | 36,457 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,229 | 22 | 36,458 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
def find(n):
if n%4==0:
return n//4
elif n%4==1:
if n>=4*2+1:
return n//4-1
else:
return -1
elif n%4==2:
if n>2:
return n//4
else:
return -1
else:
if n>=4*3+3:
return n//4-1
else:
return -1
for _ in range(int(input())):
print(find(int(input())))
``` | output | 1 | 18,229 | 22 | 36,459 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,230 | 22 | 36,460 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
n = int(input())
while n:
n-=1
num = int(input())
if num % 4 == 0:
print(num//4)
elif num % 4 == 1:
print( (num//4 - 2 + 1) if (num//4 - 2 +1) > 0 else -1 )
elif num % 4 == 2:
print( num//4 if num//4 > 0 else -1 )
elif num % 4 == 3:
print( (num//4 - 2 + 1) if (num//4 - 2 ) > 0 else -1 )
``` | output | 1 | 18,230 | 22 | 36,461 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings. | instruction | 0 | 18,231 | 22 | 36,462 |
Tags: dp, greedy, math, number theory
Correct Solution:
```
for _ in " "*int(input()):
a=int(input())
if a<4 or a in [5,7,11]:print(-1)
else:print((a//4)-((a%4)%2!=0))
``` | output | 1 | 18,231 | 22 | 36,463 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings.
Submitted Solution:
```
n = int(input())
while n:
n-=1
num = int(input())
if num % 4 == 0:
print(num//4)
elif num % 4 == 1:
print( (num//4 - 2 + 1) if (num//4 - 2 ) > 0 else -1 )
elif num % 4 == 2:
print( num//4 if num//4 > 0 else -1 )
elif num % 4 == 3:
print( (num//4 - 2 + 1) if (num//4 - 2 ) > 0 else -1 )
``` | instruction | 0 | 18,232 | 22 | 36,464 |
No | output | 1 | 18,232 | 22 | 36,465 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings.
Submitted Solution:
```
import sys
readline = sys.stdin.readline
Q = int(readline())
Ans = [None]*Q
for qu in range(Q):
N = int(readline())
Ans[qu] = N//4
if Ans[qu] == 0:
Ans[qu] = -1
print('\n'.join(map(str, Ans)))
``` | instruction | 0 | 18,233 | 22 | 36,466 |
No | output | 1 | 18,233 | 22 | 36,467 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings.
Submitted Solution:
```
n = int(input())
while n:
n-=1
num = int(input())
if num % 4 == 0:
print(num//4)
elif num % 4 == 1:
print( (num//4 - 1) if (num//4 - 1) > 0 else -1 )
elif num % 4 == 2:
print( num//4 if num//4 > 0 else -1 )
elif num % 4 == 3:
print( (num//4 - 1) if (num//4 - 1) > 0 else -1 )
``` | instruction | 0 | 18,234 | 22 | 36,468 |
No | output | 1 | 18,234 | 22 | 36,469 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings.
An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself.
Input
The first line contains single integer q (1 β€ q β€ 105) β the number of queries.
q lines follow. The (i + 1)-th line contains single integer ni (1 β€ ni β€ 109) β the i-th query.
Output
For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings.
Examples
Input
1
12
Output
3
Input
2
6
8
Output
1
2
Input
3
1
2
3
Output
-1
-1
-1
Note
12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands.
8 = 4 + 4, 6 can't be split into several composite summands.
1, 2, 3 are less than any composite number, so they do not have valid splittings.
Submitted Solution:
```
def find(n):
if n%4==0:
return n//4
elif n%4==1:
if n>=4*2+1:
return n//4-1
else:
return -1
elif n%4==2:
return n//4
else:
if n>=4*3+3:
return n//4-1
else:
return -1
for _ in range(int(input())):
print(find(int(input())))
``` | instruction | 0 | 18,235 | 22 | 36,470 |
No | output | 1 | 18,235 | 22 | 36,471 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We consider a positive integer perfect, if and only if the sum of its digits is exactly 10. Given a positive integer k, your task is to find the k-th smallest perfect positive integer.
Input
A single line with a positive integer k (1 β€ k β€ 10 000).
Output
A single number, denoting the k-th smallest perfect integer.
Examples
Input
1
Output
19
Input
2
Output
28
Note
The first perfect integer is 19 and the second one is 28. | instruction | 0 | 18,254 | 22 | 36,508 |
Tags: binary search, brute force, dp, implementation, number theory
Correct Solution:
```
def somme(n):
s=0
while(n>0):
s+=n%10
n=n//10
return s
k = int(input())
position=0
perfect=19
while(position!=k):
if(somme(perfect)==10):
position=position+1
perfect=perfect+9
else:
perfect=perfect+9
print(perfect-9)
``` | output | 1 | 18,254 | 22 | 36,509 |
Provide a correct Python 3 solution for this coding contest problem.
Find the number of integers between 1 and K (inclusive) satisfying the following condition, modulo 10^9 + 7:
* The sum of the digits in base ten is a multiple of D.
Constraints
* All values in input are integers.
* 1 \leq K < 10^{10000}
* 1 \leq D \leq 100
Input
Input is given from Standard Input in the following format:
K
D
Output
Print the number of integers satisfying the condition, modulo 10^9 + 7.
Examples
Input
30
4
Output
6
Input
1000000009
1
Output
2
Input
98765432109876543210
58
Output
635270834 | instruction | 0 | 18,356 | 22 | 36,712 |
"Correct Solution:
```
def read_int():
return int(input().strip())
def read_ints():
return list(map(int, input().strip().split(' ')))
def solve():
"""
_ _
dp[i][j][one_of_previous_number_is_lower] = count of remainder j up to i-th digit, 0 <= j < D
dp[i][(j+digit)%D][True] += dp[i-1][j][True] + dp[i-1][j][False] digit < K[i]
dp[i][(j+digit)%D][False] += dp[i-1][j][False] digit == K[i]
"""
K = list(map(int, input().strip()))
D = read_int()
dp = [[0, 0] for _ in range(D)]
dp[0][0] = 1
modulo = 10**9+7
for i in range(len(K)):
new_dp = [[0, 0] for _ in range(D)]
for j in range(D):
for digit in range(10):
new_dp[(j+digit)%D][True] += dp[j][True]
if digit < K[i]:
new_dp[(j+digit)%D][True] += dp[j][False]
if digit == K[i]:
new_dp[(j+digit)%D][False] += dp[j][False]
new_dp[(j+digit)%D][False] %= modulo
new_dp[(j+digit)%D][True] %= modulo
dp = new_dp
return (dp[0][0]+dp[0][1]-1)%modulo
if __name__ == '__main__':
print(solve())
``` | output | 1 | 18,356 | 22 | 36,713 |
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