message stringlengths 2 57.2k | message_type stringclasses 2 values | message_id int64 0 1 | conversation_id int64 61 108k | cluster float64 22 22 | __index_level_0__ int64 122 217k |
|---|---|---|---|---|---|
Provide tags and a correct Python 3 solution for this coding contest problem.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20. | instruction | 0 | 37,982 | 22 | 75,964 |
Tags: implementation, number theory
Correct Solution:
```
def main():
MOD = 2 ** 30
a, b, c = [int(i) for i in input().split()]
n = a * b * c
dp = [0 for i in range(n + 1)]
for i in range(1, n + 1):
for j in range(i, n + 1, i):
dp[j] += 1
sm = 0
for i in range(1, a + 1):
for j in range(1, b + 1):
for k in range(1, c + 1):
num = i * j * k
sm = ((sm % MOD) + (dp[num] % MOD) ) % MOD
print(sm)
if __name__ == '__main__':
main()
``` | output | 1 | 37,982 | 22 | 75,965 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
#NTFS: neharika_shah
from __future__ import division, print_function
from collections import defaultdict
mod = 1073741824
import os
import sys
from io import BytesIO, IOBase
def main():
a,b,c = map(int,input().split())
def primeFactors(n):
dic = defaultdict(int)
factors = []
while n % 2 == 0:
dic[2] += 1
n = n // 2
i = 3
while i * i <= n:
while n % i == 0:
dic[i] += 1
n = n // i
i += 2
if n > 2:
dic[n] += 1
return dic
def calc(n):
res = 1
dic = primeFactors(n)
for val in dic.values():
res = (res*(val+1))%mod
return res
# print(calc(8))
ans = 0
for i in range(1,a+1):
for j in range(1,b+1):
for k in range(1,c+1):
# print(calc(i*j*k))
ans = (ans+calc(i*j*k))%mod
print(ans)
if sys.version_info[0] < 3:
from __builtin__ import xrange as range
from future_builtins import ascii, filter, hex, map, oct, zip
"""
region fastio
Credits
template credits to cheran-senthil's github Repo
"""
BUFSIZE = 8192
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")
def print(*args, **kwargs):
"""Prints the values to a stream, or to sys.stdout by default."""
sep, file = kwargs.pop("sep", " "), kwargs.pop("file", sys.stdout)
at_start = True
for x in args:
if not at_start:
file.write(sep)
file.write(str(x))
at_start = False
file.write(kwargs.pop("end", "\n"))
if kwargs.pop("flush", False):
file.flush()
if sys.version_info[0] < 3:
sys.stdin, sys.stdout = FastIO(sys.stdin), FastIO(sys.stdout)
else:
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
if __name__ == "__main__":
main()
``` | instruction | 0 | 37,983 | 22 | 75,966 |
Yes | output | 1 | 37,983 | 22 | 75,967 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
import math
divisores = {}
def countDivisors(n) :
if n in divisores:
return divisores[n]
else:
cnt = 0
for i in range(1, (int)(math.sqrt(n)) + 1) :
if (n % i == 0) :
# If divisors are equal,
# count only one
if (n / i == i) :
cnt = cnt + 1
else : # Otherwise count both
cnt = cnt + 2
divisores[n] = cnt
return cnt
a,b,c = list(map(int, input().split(" ")))
ult_a = 1
ult_b = 1
ult_c = 1
soma = 0
for i in range(a*b*c):
ult_c = (i % c) + 1
if (i % c) == 0:
ult_b = (ult_b % b) + 1
if (i % (b*c)) == 0:
ult_a = (ult_a % a) + 1
numero = ult_a*ult_b*ult_c
numero = countDivisors(numero)
soma = (soma%1073741824) + numero
print(soma%1073741824)
``` | instruction | 0 | 37,984 | 22 | 75,968 |
Yes | output | 1 | 37,984 | 22 | 75,969 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
from math import *
def div(x):
s=0
for i in range(2,floor(sqrt(x))+1):
if x%i==0:s+=1
if x%(x//i)==0 and i!=x//i:s+=1
if x==1:return s+1
return s+2
d=[-1]*(1000001)
a,b,c=map(int,input().split());ans=0
for i in range(1,a+1):
for j in range(1,b+1):
for k in range(1,c+1):
if d[i*j*k]==-1:d[i*j*k]=div(i*j*k)
ans=(ans+d[i*j*k])
print(ans)
``` | instruction | 0 | 37,985 | 22 | 75,970 |
Yes | output | 1 | 37,985 | 22 | 75,971 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
import math as m
a, b, c=map(int, input().split())
ans=0
def div(num):
t=0
for i in range(1, int(m.sqrt(num)) + 1):
if num%i==0:
t+=1
if num//i!=i:
t+=1
return t
dic={}
for i in range(1, a+1):
for j in range(1, b+1):
for k in range(1, c+1):
a=i*j*k
if a in dic:
ans=(ans+dic[a])%1073741824
else:
dic[a]=div(a)
ans=(ans+dic[a])%1073741824
print(ans)
``` | instruction | 0 | 37,986 | 22 | 75,972 |
Yes | output | 1 | 37,986 | 22 | 75,973 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
a,b,c=[int(x) for x in input().split(' ')]
mod=1073741824
divisors=[1]*1000001
for x in range(2,1001):
for y in range(x,100001,x):
divisors[y]+=1
ans=0
for i in range(1,a+1):
for j in range(1,b+1):
for k in range(1,c+1):
ans+=(divisors[i*j*k])%mod
print(ans%mod)
``` | instruction | 0 | 37,987 | 22 | 75,974 |
No | output | 1 | 37,987 | 22 | 75,975 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
a,b,c = map(int,input().split())
li = a*b*c +1
d = [1]*li
for i in range(2,li):
for j in range(i,li,i):
d[j]+=1
ans = 0
for i in range(1,a+1):
for j in range(1,b+1):
for z in range(1,c+1):
ans += d[a*b*c]
print(ans)
``` | instruction | 0 | 37,988 | 22 | 75,976 |
No | output | 1 | 37,988 | 22 | 75,977 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
from math import sqrt, ceil
a, b, c = list(map(int, input().split()))
N = (a * b * c) + 1
def generate_prime(N, pp, prime):
pp[1] = False
for i in range(2, N):
if pp[i]:
prime.append(i)
x = i ** 2
while x < N:
pp[x] = False
x = x + i
prime = []
pp = [True] * N
generate_prime(N, pp, prime)
def divisor_count(num):
ans = 1
if num == 1:
return 1
for p in prime:
factor = 0
while num:
if num % p == 0:
factor += 1
num = num // p
else:
ans = ans * (factor + 1)
break
if pp[num]:
ans = ans * 2
break
if num == 1:
break
return ans
d = {}
res = 0
for i in range(1, a+1):
for j in range(1, b+1):
for k in range(1, c+1):
t = i * j * k
if t not in d:
d[t] = divisor_count(t)
res = res + d[t]
``` | instruction | 0 | 37,989 | 22 | 75,978 |
No | output | 1 | 37,989 | 22 | 75,979 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Let's denote d(n) as the number of divisors of a positive integer n. You are given three integers a, b and c. Your task is to calculate the following sum:
<image>
Find the sum modulo 1073741824 (230).
Input
The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 100).
Output
Print a single integer — the required sum modulo 1073741824 (230).
Examples
Input
2 2 2
Output
20
Input
5 6 7
Output
1520
Note
For the first example.
* d(1·1·1) = d(1) = 1;
* d(1·1·2) = d(2) = 2;
* d(1·2·1) = d(2) = 2;
* d(1·2·2) = d(4) = 3;
* d(2·1·1) = d(2) = 2;
* d(2·1·2) = d(4) = 3;
* d(2·2·1) = d(4) = 3;
* d(2·2·2) = d(8) = 4.
So the result is 1 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 20.
Submitted Solution:
```
#!/usr/bin/env python3
import sys
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
dp = [[0,1]]*(101)
dp[1][0] == 1
def factors(n):
if dp[n][0] == 1:
return dp[n][1]
s = 1
for i in range(len(primes)):
if (n%primes[i] == 0):
c = 0
while (n% primes[i] ==0):
c += 1
n = n//primes[i]
s *= c+1
dp[n][0] = 1
dp[n][1] = s
return s
if __name__ == "__main__":
a, b, c = map(int, sys.stdin.readline().split())
s = 0
for i in range(1, a+1):
for j in range(1, b+1):
for k in range(1, c+1):
s += (factors(i)+1)*(factors(j)+1)*(factors(k)+1)
print(s%1073741824)
``` | instruction | 0 | 37,990 | 22 | 75,980 |
No | output | 1 | 37,990 | 22 | 75,981 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,065 | 22 | 76,130 |
Tags: math, number theory
Correct Solution:
```
a, b = map(int, input().split())
k, val = 1, 0
while k * k <= a - b:
if (a - b) % k == 0:
val += sum(1 for x in {k, (a - b) // k} if x > b)
k += 1
print('infinity' if a == b else val)
``` | output | 1 | 38,065 | 22 | 76,131 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,066 | 22 | 76,132 |
Tags: math, number theory
Correct Solution:
```
from math import *
a, b = map( int, input().split() )
if( a < b ):
print( 0 )
elif( a == b ):
print( "infinity" )
else:
a -= b
cnt = 0
last = int( sqrt( a ) )
for i in range( 1, ( last + 1 ) ):
if( a % i == 0 ):
if( i > b ):
cnt += 1
if( ( a//i > b ) and ( a//i > i ) ):
cnt += 1
print( cnt )
``` | output | 1 | 38,066 | 22 | 76,133 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,067 | 22 | 76,134 |
Tags: math, number theory
Correct Solution:
```
arr = [int(x) for x in input().split()]
a = arr[0]
b = arr[1]
resp = 0
if a == b:
resp = 'infinity'
if resp != 'infinity':
x = a - b
i = 1
c = 0
while i**2 < x:
c += 1
if x % i == 0:
if i > b:
resp += 1
if x/i > b:
resp += 1
i += 1
if i**2 == x and i > b:
resp += 1
print (resp)
``` | output | 1 | 38,067 | 22 | 76,135 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,068 | 22 | 76,136 |
Tags: math, number theory
Correct Solution:
```
a=input().split()
n=int(a[0]);m=int(a[1])
ans=0
if n<m :
print('0')
else :
if n==m:
print("infinity")
else :
i=1
n=n-m
while i*i<n:
if n%i==0:
if i>m:
ans=ans+1
if n/i>m:
ans=ans+1
i=i+1
s=int(n**0.5)
if s**2==n and s>m :
ans=ans+1
print(ans)
``` | output | 1 | 38,068 | 22 | 76,137 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,069 | 22 | 76,138 |
Tags: math, number theory
Correct Solution:
```
a, b = list(map(int,input().split()))
if a == b:
print("infinity")
exit(0)
a, ans = a - b, 0
for i in range(1,a+1):
if i * i > a: break
if a % i != 0: continue
if i > b: ans += 1
if i != a // i and a // i > b: ans += 1
print(ans)
``` | output | 1 | 38,069 | 22 | 76,139 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,070 | 22 | 76,140 |
Tags: math, number theory
Correct Solution:
```
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
import time
#
(a, b) = (int(i) for i in input().split())
start = time.time()
ans = 0
if ( a == b):
ans = "infinity"
elif (a > b):
i = 1
max = (a-b) ** 0.5
while(i <= max ):
if (divmod(a-b, i)[1] == 0):
if i > b:
ans += 1
buf = (a-b)//i
if buf > b and buf != i:
ans += 1
i += 1
print(ans)
finish = time.time()
#print(finish - start)
``` | output | 1 | 38,070 | 22 | 76,141 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,071 | 22 | 76,142 |
Tags: math, number theory
Correct Solution:
```
import sys
def fastio():
from io import StringIO
from atexit import register
global input
sys.stdin = StringIO(sys.stdin.read())
input = lambda : sys.stdin.readline().rstrip('\r\n')
sys.stdout = StringIO()
register(lambda : sys.__stdout__.write(sys.stdout.getvalue()))
fastio()
def debug(*var, sep = ' ', end = '\n'):
print(*var, file=sys.stderr, end = end, sep = sep)
INF = 10**20
MOD = 10**9 + 7
I = lambda:list(map(int,input().split()))
a,b = list(map(int , input().split()))
if(a<b): print(0)
else:
if a==b:
print('infinity')
elif a>b:
cnt =0
for i in range(1,int((a-b)**0.5)+1):
if(a-b)%i==0:
if(i>b): cnt+=1
if(a-b)/i > b and i*i != (a-b) : cnt+=1
print(cnt)
``` | output | 1 | 38,071 | 22 | 76,143 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image> | instruction | 0 | 38,072 | 22 | 76,144 |
Tags: math, number theory
Correct Solution:
```
def printDivisors(n) :
ans = []
for i in range(1, int(n ** 0.5 + 1)):
if n % i == 0:
if n // i == i:
ans.append(i)
else:
ans.append(i)
ans.append(n//i)
return sorted(ans)
a, b = map(int, input().split())
print('infinity') if a == b else print(0) if a < b else print(sum(i > b for i in printDivisors(a - b)))
``` | output | 1 | 38,072 | 22 | 76,145 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import math
def funcao(a, b):
if a == b:
return -1
if a == 0:
if b == 0:
return - 1
else:
return 0
if a < b:
return 0
divisor = 1
cont = 0
while divisor <= math.sqrt(a - b):
quociente = (a - b) / divisor
if (a - b) % divisor == 0:
if divisor > b:
cont += 1
if quociente != divisor and quociente > b:
cont += 1
divisor += 1
return cont
entrada = input().split()
a = int(entrada[0])
b = int(entrada[1])
res = funcao(a, b)
if res == -1:
print("infinity")
else:
print(res)
``` | instruction | 0 | 38,073 | 22 | 76,146 |
Yes | output | 1 | 38,073 | 22 | 76,147 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import itertools
import math
def main():
a, b = list(map(int, input().split()))
if a == b:
print("infinity")
return
if a < b:
print(0)
return
a -= b
n = 0
for i in range(1, math.ceil(math.sqrt(a))):
if (a % i == 0):
n += (i > b) + (a // i > b)
if math.sqrt(a) % 1 == 0 and math.sqrt(a) > b:
n += 1
print(n)
if __name__ == "__main__":
main()
``` | instruction | 0 | 38,074 | 22 | 76,148 |
Yes | output | 1 | 38,074 | 22 | 76,149 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import sys
a, b = map(int, input().split())
dis = a - b
if dis == 0:
print("infinity")
else:
res = 0
x = 1
while x ** 2 <= dis:
if dis % x == 0 and x > b:
res += 1
if dis % x == 0 and dis // x > b and x ** 2 != dis:
res += 1
x += 1
print(res)
``` | instruction | 0 | 38,075 | 22 | 76,150 |
Yes | output | 1 | 38,075 | 22 | 76,151 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import math
primeiro,segundo = map(int, input().split())
diferenca = primeiro-segundo
saida = 0
if(primeiro == segundo):
print("infinity")
elif(primeiro < segundo):
print('0')
else:
limite = int(math.sqrt(diferenca))+1
for numero in range(1,limite):
if(diferenca % numero == 0):
if((numero*numero) != diferenca):
if((diferenca/numero)>segundo):
saida += 1
if(numero>segundo):
saida += 1
else:
if(numero>segundo):
saida += 1
print(saida)
``` | instruction | 0 | 38,076 | 22 | 76,152 |
Yes | output | 1 | 38,076 | 22 | 76,153 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import bisect
from itertools import accumulate, count
import os
import sys
import math
from decimal import *
from io import BytesIO, IOBase
BUFSIZE = 8192
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)
def input(): return sys.stdin.readline().rstrip("\r\n")
def isPrime(n) :
if (n <= 1) : return False
if (n <= 3) : return True
if (n % 2 == 0 or n % 3 == 0) : return False
i = 5
while(i * i <= n) :
if (n % i == 0 or n % (i + 2) == 0) :
return False
i = i + 6
return True
def SieveOfEratosthenes(n):
prime=[]
primes = [True for i in range(n+1)]
p = 2
while (p * p <= n):
if (primes[p] == True):
for i in range(p * p, n+1, p):
primes[i] = False
p += 1
for i in range(2,len(primes)):
if primes[i]==True:
prime.append(i)
return prime
def primefactors(n):
fac=[]
while(n%2==0):
fac.append(2)
n=n//2
for i in range(3,int(math.sqrt(n))+2):
while(n%i==0):
fac.append(i)
n=n//i
if n>1:
fac.append(n)
return fac
def factors(n):
fac=set()
fac.add(1)
fac.add(n)
for i in range(2,int(math.sqrt(n))+1):
if n%i==0:
fac.add(i)
fac.add(n//i)
return list(fac)
def NcR(n, r):
p = 1
k = 1
if (n - r < r):
r = n - r
if (r != 0):
while (r):
p *= n
k *= r
m = math.gcd(p, k)
p //= m
k //= m
n -= 1
r -= 1
else:
p = 1
return p
def Log2(x):
if x == 0:
return False;
return (math.log10(x) /
math.log10(2));
def isPowerOfTwo(n):
return (math.ceil(Log2(n)) ==
math.floor(Log2(n)));
#--------------------------------------------------------codeby apurva3455/AD18
a,b=map(int,input().split())
if a==0:
print(0)
elif b>=a:
print("infinity")
else:
c=a-b
fac=factors(c)
count=0
for i in range(0,len(fac)):
if fac[i]>=b:
count+=1
print(count)
``` | instruction | 0 | 38,077 | 22 | 76,154 |
No | output | 1 | 38,077 | 22 | 76,155 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
import math
def funcao(a, b):
if a == b:
return -1
divisor = 1
cont = 0
while divisor <= math.sqrt(a - b):
quociente = (a - b) / divisor
if (a - b) % divisor == 0:
if divisor > b:
cont += 1
if quociente != divisor and quociente > b:
cont += 1
divisor += 1
return cont
entrada = input().split()
a = int(entrada[0])
b = int(entrada[1])
print(funcao(a, b))
``` | instruction | 0 | 38,078 | 22 | 76,156 |
No | output | 1 | 38,078 | 22 | 76,157 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
a, b = map(int, input().split())
if b > a:
print(0)
elif a == b:
print("infinity")
else:
ds = []
r = int((a-b)**0.5)
for d in range(1, r):
if (a-b)%d == 0:
ds.append(d)
ds.append((a-b)//d)
if (a-b) == r*r:
ds.append(r)
print(len([d for d in ds if d > b]))
``` | instruction | 0 | 38,079 | 22 | 76,158 |
No | output | 1 | 38,079 | 22 | 76,159 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Last week, Hamed learned about a new type of equations in his math class called Modular Equations. Lets define i modulo j as the remainder of division of i by j and denote it by <image>. A Modular Equation, as Hamed's teacher described, is an equation of the form <image> in which a and b are two non-negative integers and x is a variable. We call a positive integer x for which <image> a solution of our equation.
Hamed didn't pay much attention to the class since he was watching a movie. He only managed to understand the definitions of these equations.
Now he wants to write his math exercises but since he has no idea how to do that, he asked you for help. He has told you all he knows about Modular Equations and asked you to write a program which given two numbers a and b determines how many answers the Modular Equation <image> has.
Input
In the only line of the input two space-separated integers a and b (0 ≤ a, b ≤ 109) are given.
Output
If there is an infinite number of answers to our equation, print "infinity" (without the quotes). Otherwise print the number of solutions of the Modular Equation <image>.
Examples
Input
21 5
Output
2
Input
9435152 272
Output
282
Input
10 10
Output
infinity
Note
In the first sample the answers of the Modular Equation are 8 and 16 since <image>
Submitted Solution:
```
a,b=input().split()
a=int(a)
b=int(b)
sel=a-b
x=0
i=1
while((sel/i)>b):
if(sel//i)==(sel/i):
x+=1
i+=1
print("infinity" if x==0 else x)
``` | instruction | 0 | 38,080 | 22 | 76,160 |
No | output | 1 | 38,080 | 22 | 76,161 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,111 | 22 | 76,222 |
Tags: math, number theory
Correct Solution:
```
mod_number = 10 ** 9 + 7
def power_mod(n, p):
if p < 2:
return (n ** p) % mod_number
sub_result = power_mod(n, p // 2)
if p % 2 == 0:
return (sub_result ** 2) % mod_number
else:
return (sub_result ** 2 * n) % mod_number
def get_frequency_map(items):
frequency_map = {}
for item in items:
if item in frequency_map:
frequency_map[item] += 1
else:
frequency_map[item] = 1
return frequency_map
def get_product_of_others(items):
length = len(items)
prefix_product_of_others = [1] * length
suffix_product_of_others = [1] * length
for i in range(1, length):
prefix_product_of_others[i] = (prefix_product_of_others[i - 1] * items[i - 1]) % (mod_number - 1)
for i in reversed(range(length - 1)):
suffix_product_of_others[i] = (suffix_product_of_others[i + 1] * items[i + 1]) % (mod_number - 1)
return [prefix_product_of_others[i] * suffix_product_of_others[i]
for i in range(length)]
def main():
m = int(input())
prime_factors = [int(t) for t in input().split()]
prime_factors_count_map = get_frequency_map(prime_factors)
ordered_prime_factors = list(prime_factors_count_map.keys())
# if prime factor 2 occurs three times
# it can be choose four way to form other factors
# taking three times, taking two times, taking one times and not taking 2 to form a factor
each_prime_factor_choices = [prime_factors_count_map[prime_factor] + 1 for prime_factor in ordered_prime_factors]
other_prime_factors_choices = get_product_of_others(each_prime_factor_choices)
total_factors = 1
for i, prime_factor in enumerate(ordered_prime_factors):
prime_factor_count = prime_factors_count_map[prime_factor]
# if prime factor 2 occurs four time then possible factors by 2, 4, 8
# means product of the factors will be 2^(1+2+3)
total_power_of_factor = ((prime_factor_count * (prime_factor_count + 1)) // 2)
product_of_factor_by_prime_factor = power_mod(prime_factor, total_power_of_factor % (mod_number - 1))
total_factors *= power_mod(product_of_factor_by_prime_factor, other_prime_factors_choices[i] % (mod_number - 1))
print(total_factors % mod_number)
if __name__ == '__main__':
main()
``` | output | 1 | 38,111 | 22 | 76,223 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,112 | 22 | 76,224 |
Tags: math, number theory
Correct Solution:
```
MOD = 10 ** 9 + 7
ans = 1
tau = 1
N = 200005
cnt = [0] * N
n = int(input())
for x in input().split() :
cnt[int(x)] += 1
for i in range(1, N) :
if cnt[i] == 0 : continue
ans = pow (ans, cnt[i] + 1, MOD) * pow (i, cnt[i] * (cnt[i] + 1) // 2 % (MOD - 1) * tau % (MOD - 1), MOD) % MOD
tau = tau * (cnt[i] + 1) % (MOD - 1)
print (ans)
``` | output | 1 | 38,112 | 22 | 76,225 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,113 | 22 | 76,226 |
Tags: math, number theory
Correct Solution:
```
from functools import reduce
import sys
#f = open('test', 'r')
#sys.stdin = f
m = int(input())
p = 10**9 + 7
dividors = [int(x) for x in input().split()]
D = {}
for d in dividors:
if d in D:
D[d] += 1
else:
D[d] = 1
prod = reduce(lambda x,y : x*y%p, dividors)
deg = reduce(lambda x,y : x*y, (d+1 for d in D.values()))
if deg % 2:
prod = reduce(lambda x,y : x*y%p, (pow(d, i//2, p) for d,i in D.items()))
ans = pow(prod, deg%(p-1), p)
else:
ans = pow(prod, (deg//2)%(p-1), p)
print(ans)
``` | output | 1 | 38,113 | 22 | 76,227 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,114 | 22 | 76,228 |
Tags: math, number theory
Correct Solution:
```
M = 10 ** 9 + 7
m = int(input())
A = list(map(int, input().split()))
D = {}
for i in A:
if i not in D:
D[i] = 0
D[i] += 1
alph = 1
for j in D:
alph *= (D[j] + 1)
ans = 1
for j in D:
ans *= pow(j, alph * D[j] // 2 % (M - 1), M)
ans %= M
print(ans)
``` | output | 1 | 38,114 | 22 | 76,229 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,115 | 22 | 76,230 |
Tags: math, number theory
Correct Solution:
```
MD = 1000000007
m = int(input())
p = list(map(int, input().split()))
q = {}
for el in p:
if el in q:
q[el] += 1
else:
q[el] = 2
sum1 = 1
sum2 = 1
for el in q:
sum1=sum1*q[el]
sum2=sum2*pow(el,(q[el]-1),MD)
sum=pow(sum2,sum1//2,MD)
if sum1 % 2 == 1:
for el in q:
for i in range(q[el]//2):
sum = (sum * el) % MD
print(sum)
``` | output | 1 | 38,115 | 22 | 76,231 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,116 | 22 | 76,232 |
Tags: math, number theory
Correct Solution:
```
m = int(input())
primes = list(map(int, input().split()))
dict = {}
result = 1
for p in primes:
dict[p] = dict.get(p, 0) + 1
mult = 1
for x in dict.values():
mult *= x + 1
mult %= 2*(10**9+6)
for x, y in dict.items():
result *= pow(x, (y*mult)//2, 10**9 + 7)
result %= 10**9 + 7
print(result)
``` | output | 1 | 38,116 | 22 | 76,233 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,117 | 22 | 76,234 |
Tags: math, number theory
Correct Solution:
```
from collections import Counter
M = 10**9 + 7
def multipliers(pfs):
y = 1
for c in pfs.values():
y *= c+1
if y % 2:
sx = 1
for p, c in pfs.items():
sx = (sx * pow(p, c//2, M)) % M
return pow(sx, y, M)
else:
x = 1
for p, c in pfs.items():
x = (x * pow(p, c, M)) % M
return pow(x, y//2, M)
if __name__ == '__main__':
m = int(input())
pfs = Counter(map(int, input().split()))
print(multipliers(pfs))
``` | output | 1 | 38,117 | 22 | 76,235 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728. | instruction | 0 | 38,118 | 22 | 76,236 |
Tags: math, number theory
Correct Solution:
```
import sys
#Library Info(ACL for Python/Pypy) -> https://github.com/not522/ac-library-python
def input():
return sys.stdin.readline().rstrip()
DXY = [(0, -1), (1,0), (0, 1), (-1,0)] #L,D,R,Uの順番
mod = int(1e9 + 7)
from collections import defaultdict
def main():
n = int(input())
d = defaultdict(int)
a = list(map(int,input().split()))
for v in a:d[v] += 1
group_product = 1
group_size = 1 # modulo mod - 1
for p,ord in d.items():
group_product = pow(group_product,ord + 1,mod)
group_product *= pow(p,ord * (ord + 1) // 2 * group_size,mod)
group_product %= mod
group_size *= (ord + 1)
group_size %= (mod - 1)
print(group_product % mod)
return 0
if __name__ == "__main__":
main()
``` | output | 1 | 38,118 | 22 | 76,237 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
mod=10**9+7
def pow(a,b): #Algoritmo de Exponenciacion binaria
exp=1 # Caso base a^1=a
x=a % mod
b=b%(mod-1)
while b > 0:
if b % 2 == 1:# En caso que b sea impar
exp=(exp*x)%mod
# En caso que b sea impar
x=(x*x)%mod
b=b//2
return exp%mod
x=input() #
primes=map(int,input().split())# Capturo la entrada del problema
primesDicc=dict() # inicializo un diccionario para guardar la descomposicion en primos de la forma <pi,e1>donde e1 es el exponente con que aparece pi en la descomposicion
n=1;
for p in primes:
n=(n*p)%mod
if p not in primesDicc:
primesDicc[p]=1
else:
primesDicc[p]+=1
# Mapping(primes)
countdiv =1
raiz=1
for p in primesDicc: #Hago un recorrido por todos las llaves de mi diccionario para generar el numero de divisores
countdiv *= primesDicc[p]+1
result=0
if countdiv % 2 == 0:
result=pow(n,countdiv//2)
else:
for p in primesDicc:
raiz=(raiz*pow(p,primesDicc[p]//2))
result=(raiz*pow(n,countdiv//2))%mod
print(result)
``` | instruction | 0 | 38,119 | 22 | 76,238 |
Yes | output | 1 | 38,119 | 22 | 76,239 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
n = int(input())
exps = {}
for x in [ int(x) for x in input().split() ]:
exps[x] = exps.get(x,0)+1
r,m = 1,1
M = 1000000007
P = M-1
for p,e in exps.items():
E = (e*(e+1)//2)%P
E = E*m%P
r = pow(r,e+1,M)*pow(p,E,M)%M
m = m*(e+1)%P
print(r)
# C:\Users\Usuario\HOME2\Programacion\ACM
``` | instruction | 0 | 38,120 | 22 | 76,240 |
Yes | output | 1 | 38,120 | 22 | 76,241 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
import math
m=10**9+7
n = int(input())
lis = list(map(int,input().split()))
ans=[0]*(200005)
t=ta=1
for i in lis:
ans[i]+=1
for i in range(1,200005):
if ans[i]==0:
continue
else:
p=pow(i,ans[i]*(ans[i]+1)//2 % (m-1),m)
t=pow(t, ans[i]+1 ,m)*pow(p,ta,m) % m
ta=(ta*(ans[i]+1))%(m-1)
print(t)
``` | instruction | 0 | 38,121 | 22 | 76,242 |
Yes | output | 1 | 38,121 | 22 | 76,243 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
MD = 1000000007
m = int(input())
p = list(map(int, input().split()))
q = {}
for el in p:
if el in q:
q[el] += 1
else:
q[el] = 2
sum1 = 1
sum2 = 1
for el in q:
sum1=sum1*q[el]
sum2=sum2*pow(el,(q[el]-1),MD)
sum=pow(sum2,sum1//2,MD)
if sum1 % 2 == 1:
for el in q:
sum = (sum * pow(el,q[el]//2,MD)) % MD
print(sum)
``` | instruction | 0 | 38,122 | 22 | 76,244 |
Yes | output | 1 | 38,122 | 22 | 76,245 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
M = 1000000007
def mpow (n, p):
if p == 0:
return 1
ans = mpow(n, int(p/2))
ans = (ans * ans) % M
if p % 2 == 1:
ans = ans * n
return ans % M
n = int(input())
arr = input().split(' ')
m = [0] * 200001
for i in arr:
m[int(i)] = m[int(i)] + 1
d = 1
for i in m:
d = d * (i + 1)
ans = 1
e = -1
for i in m:
e = e + 1
if i == 0:
continue
ans = (ans * mpow(e, (i * d)/2)) % M
print (ans % M)
``` | instruction | 0 | 38,123 | 22 | 76,246 |
No | output | 1 | 38,123 | 22 | 76,247 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
m = int(input())
a = list(map(int, input().split()))
pa = {}
for p in a:
if p not in pa:
pa[p] = 0
pa[p] += 1
M = 10 ** 9 + 7
G = 1
for p in pa:
G *= (pa[p] + 1)
G %= M
ans = 1
for p in pa:
x = G
x *= (pa[p] * (pa[p] + 1) // 2) * pow(pa[p] + 1, M - 2, M) % M
x %= M
ans *= pow(p, x, M)
ans %= M
print(ans)
``` | instruction | 0 | 38,124 | 22 | 76,248 |
No | output | 1 | 38,124 | 22 | 76,249 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
import math
m = int(input())
n = 1
u = 1
a = list(map(int, input().split()))
for i in range(len(a)):
n *= a[i]
for i in range(1, int(math.sqrt(n)) + 1, 1):
if int(n / i) * int(i) == n:
u *= i % 1000000007;
if int(i) * int(n / i) == n and i * i != n:
u *= (n / i) % 1000000007
print(i)
u %= 1000000007
print(int(u))
``` | instruction | 0 | 38,125 | 22 | 76,250 |
No | output | 1 | 38,125 | 22 | 76,251 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ayrat has number n, represented as it's prime factorization pi of size m, i.e. n = p1·p2·...·pm. Ayrat got secret information that that the product of all divisors of n taken modulo 109 + 7 is the password to the secret data base. Now he wants to calculate this value.
Input
The first line of the input contains a single integer m (1 ≤ m ≤ 200 000) — the number of primes in factorization of n.
The second line contains m primes numbers pi (2 ≤ pi ≤ 200 000).
Output
Print one integer — the product of all divisors of n modulo 109 + 7.
Examples
Input
2
2 3
Output
36
Input
3
2 3 2
Output
1728
Note
In the first sample n = 2·3 = 6. The divisors of 6 are 1, 2, 3 and 6, their product is equal to 1·2·3·6 = 36.
In the second sample 2·3·2 = 12. The divisors of 12 are 1, 2, 3, 4, 6 and 12. 1·2·3·4·6·12 = 1728.
Submitted Solution:
```
from functools import reduce
import operator
from collections import Counter
import copy
n = int(input())
arr = [int(i) for i in input().split()]
maxnum = reduce(operator.mul, arr, 1)
cnt = Counter(arr)
for k, v in cnt.items():
t = k
while v > 1:
t *= k
arr.append(t)
v -= 1
table = set(arr)
data = list(table)
for a in arr:
for d in data:
if max(d, a) % min(d, a) != 0:
t = d * a
if t <= maxnum and t not in table:
table.add(t)
data.append(t)
print(reduce(operator.mul, data, 1))
``` | instruction | 0 | 38,126 | 22 | 76,252 |
No | output | 1 | 38,126 | 22 | 76,253 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,714 | 22 | 77,428 |
Tags: math, number theory
Correct Solution:
```
from functools import reduce
def factors(n):
return set(reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0)))
for _ in range(int(input())):
N = int(input())
List = [int(x) for x in input().split()]
List.sort()
Ans = List[0]*List[-1]
flag = 0
for i in range((N>>1) + 1):
if(List[i] * List[N-i-1] != Ans):
flag = 1
break
if(flag):
print(-1)
else:
given = set(List)
given.add(1)
given.add(Ans)
if(factors(Ans) == given):
print(Ans)
else:
print(-1)
``` | output | 1 | 38,714 | 22 | 77,429 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,715 | 22 | 77,430 |
Tags: math, number theory
Correct Solution:
```
import math
t = int(input())
for i in range(t):
n = int(input())
a = list(map(int,input().split()))
a.sort()
qw = []
qwqw = []
c = 0
for j in range(math.ceil(len(a) / 2)):
qw.append(a[j] * a[len(a) - j - 1])
if qw.count(qw[0]) == len(qw):
for j in range(2,int(math.sqrt(qw[0])) + 1):
if qw[0] % j == 0:
qwqw.append(j)
if j != qw[0] // j:
qwqw.append(qw[0] // j)
qwqw.sort()
if len(a) == len(qwqw):
for j in range(len(a)):
if a[j] != qwqw[j]:
c = 1
break
if c == 0:
print(qw[0])
else:
print(-1)
else:
print(-1)
else:
print(-1)
``` | output | 1 | 38,715 | 22 | 77,431 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,716 | 22 | 77,432 |
Tags: math, number theory
Correct Solution:
```
t=int(input())
def prov1(a):
pp=a[0]*a[-1]
for i in range(len(a)):
if a[i]*a[-1-i]!=pp: return -1
return pp
def kkk(a):
i=2
d=0
while i*i<a:
if a%i==0:
d+=2
i+=1
if i*i==a: d+=1
return d
def koka(a,b):
ans=0
while a%b==0:
a=a//b
ans+=1
return ans
for i in range(t):
n=int(input())
a=sorted(list(map(int,input().split())))
y=prov1(a)
if y==-1:
print(-1)
else:
yy=kkk(a[0])
if yy>0: print(-1)
else:
u=kkk(a[-1])+2
z=koka(a[-1],a[0])
if z==0: d=u*2-2
else:
d=(u//(z+1))*(z+2)-2
if d==n:
print(y)
else:
print(-1)
``` | output | 1 | 38,716 | 22 | 77,433 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,717 | 22 | 77,434 |
Tags: math, number theory
Correct Solution:
```
def dl(x):
d = 2
new = []
while d * d < x:
if x % d == 0:
new.append(d)
new.append(x // d)
d += 1
if d * d == x:
new.append(d)
return new
t = int(input())
for i in range(t):
n = int(input())
sp = list(map(int, input().split()))
sp.sort()
ch = sp[0] * sp[-1]
new = dl(ch)
new.sort()
if sp == new:
print(ch)
else:
print(-1)
``` | output | 1 | 38,717 | 22 | 77,435 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,718 | 22 | 77,436 |
Tags: math, number theory
Correct Solution:
```
import math
def getFirstSetBitPos(n):
return math.log2(n & -n) + 1
def find_div(x):
ls=[]
for i in range(2,int(x**0.5)+1):
if x%i==0:
ls.append(i)
if i!=x//i:
ls.append(x//i)
return sorted(ls)
for _ in range(int(input())):
n = int(input())
arr = list(map(int, input().split()))
arr.sort()
ans=arr[0]*arr[-1]
if find_div(ans)==arr:
print(ans)
else:
print(-1)
``` | output | 1 | 38,718 | 22 | 77,437 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,719 | 22 | 77,438 |
Tags: math, number theory
Correct Solution:
```
def ans(arr):
i,j = 0,len(arr)-1
k = arr[i]*arr[j]
while i<=j:
if arr[i]*arr[j] != k:
return -1
i += 1
j -= 1
li =[]
i = 2
while i*i<=k:
if k%i==0:
li.append(i)
if i*i != k:
li.append(k//i)
i+=1
if not (sorted(li)==arr):
return -1
return k
for _ in range(int(input())):
n = int(input())
arr = [int(s) for s in input().split()]
arr.sort()
print(ans(arr))
``` | output | 1 | 38,719 | 22 | 77,439 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,720 | 22 | 77,440 |
Tags: math, number theory
Correct Solution:
```
for _ in range(int(input())):
n=int(input())
a=list(map(int,input().split()))
x=min(a)*max(a)
l=set()
d=2
while d*d<=x:
if not x%d:
l.add(d)
l.add(x//d)
d+=1
print(x if l==set(a) else -1)
``` | output | 1 | 38,720 | 22 | 77,441 |
Provide tags and a correct Python 3 solution for this coding contest problem.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4 | instruction | 0 | 38,721 | 22 | 77,442 |
Tags: math, number theory
Correct Solution:
```
from random import randint
from math import gcd
def pal(x):
p=False
a = randint(2, x - 1)
if gcd(a,x)==1:
if a**(x-1)%x==1:
p=True
else:
return False
return p
t = int(input())
for q in range(t):
n = int(input())
dn = input().split(' ')
d = [int(n) for n in dn]
d.sort()
p = d[-1] * d[0]
ct = 0
neg1=False
if n==1 and d[0]>=10**3 :
print(p)
neg1=True
elif n<=6:
for i in range(2, p):
if p % i == 0:
ct += 1
if ct > n:
print(-1)
neg1 = True
break
elif ct==0 and i==5:
break
if pal(d[0]):
print(-1)
neg=True
break
if not(neg1):
for i in range(n//2 +1):
if not (d[i] * d[-1 - i] == p):
neg1=True
print(-1)
break
i=0
if not(neg1):
while i < len(d):
if d[i]!=d[-1] and d[i]!=d[0] and not (p % d[i] == 0):
i = 0
d = d[:-1]
p = d[-1] * d[0]
i+=1
print(p)
``` | output | 1 | 38,721 | 22 | 77,443 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
We guessed some integer number x. You are given a list of almost all its divisors. Almost all means that there are all divisors except 1 and x in the list.
Your task is to find the minimum possible integer x that can be the guessed number, or say that the input data is contradictory and it is impossible to find such number.
You have to answer t independent queries.
Input
The first line of the input contains one integer t (1 ≤ t ≤ 25) — the number of queries. Then t queries follow.
The first line of the query contains one integer n (1 ≤ n ≤ 300) — the number of divisors in the list.
The second line of the query contains n integers d_1, d_2, ..., d_n (2 ≤ d_i ≤ 10^6), where d_i is the i-th divisor of the guessed number. It is guaranteed that all values d_i are distinct.
Output
For each query print the answer to it.
If the input data in the query is contradictory and it is impossible to find such number x that the given list of divisors is the list of almost all its divisors, print -1. Otherwise print the minimum possible x.
Example
Input
2
8
8 2 12 6 4 24 16 3
1
2
Output
48
4
Submitted Solution:
```
t = int(input())
for q in range(t):
n = int(input())
a = list(map(int, input().split()))
ans = min(a)*max(a)
deliver = [q for q in range(2, round(ans**(1/2))+1) if ans % q == 0]
if deliver+[ans//deliver[q] for q in range(len(deliver)-1-(deliver[-1]**2 == ans), -1, -1)] == sorted(a):
print(ans)
else:
print(-1)
``` | instruction | 0 | 38,722 | 22 | 77,444 |
Yes | output | 1 | 38,722 | 22 | 77,445 |
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