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.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,506 | 22 | 57,012 |
Tags: math
Correct Solution:
```
def solve(A):
odd = sum([1 for it in A if it%2])
return "Yes" if 2*odd == len(A) else "No"
for case in range(int(input())):n = int(input());print(solve(list(map(int, input().split()))))
``` | output | 1 | 28,506 | 22 | 57,013 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,507 | 22 | 57,014 |
Tags: math
Correct Solution:
```
m = int(input())
for i in range(m):
n = int(input())
lis = list(map(int, input().split()))
odd = 0
even = 0
for j in lis:
if(j % 2 == 0):
even += 1
else:
odd += 1
if(odd == even):
print("Yes")
else:
print("No")
``` | output | 1 | 28,507 | 22 | 57,015 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,508 | 22 | 57,016 |
Tags: math
Correct Solution:
```
n = int(input())
for _ in range(n):
n_array = int(input())
array = list(map(int, input().split()))
if len([1 for elem in array if elem % 2 == 0]) == len([1 for elem in array if elem % 2 == 1]):
print('Yes')
else:
print('No')
``` | output | 1 | 28,508 | 22 | 57,017 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,509 | 22 | 57,018 |
Tags: math
Correct Solution:
```
t=int(input())
for _ in range(0,t):
n=int(input())
a=list(map(int,input().split()))
ce=0
co=0
for i in a:
if(i%2==0):
ce=ce+1
if(i%2==1):
co=co+1
if(ce==co):
print("Yes")
else:
print("No")
``` | output | 1 | 28,509 | 22 | 57,019 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,510 | 22 | 57,020 |
Tags: math
Correct Solution:
```
num_lines = int(input())
def solve(nums):
count_ev = 0
count_od = 0
for num in nums:
if num % 2 == 0:
count_ev += 1
else:
count_od += 1
if count_od == count_ev:
return True
else:
return False
i = 0
while i < num_lines:
n = int(input ())
nums = [int(x) for x in input().split()]
if solve(nums):
print('Yes')
else:
print('No')
i+=1
``` | output | 1 | 28,510 | 22 | 57,021 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,511 | 22 | 57,022 |
Tags: math
Correct Solution:
```
import sys
input = lambda:sys.stdin.readline()
int_arr = lambda: list(map(int,input().split()))
str_arr = lambda: list(map(str,input().split()))
get_str = lambda: map(str,input().split())
get_int = lambda: map(int,input().split())
get_flo = lambda: map(float,input().split())
mod = 1000000007
def solve(n,arr):
o = 0
for i in arr:
if i % 2 != 0:
o += 1
if o % n == 0 and (o//n) % 2 != 0:
print("Yes")
else:
print("No")
for _ in range(int(input())):
n = int(input())
arr = int_arr()
solve(n,arr)
``` | output | 1 | 28,511 | 22 | 57,023 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,512 | 22 | 57,024 |
Tags: math
Correct Solution:
```
def f(l):
x=0
for i in range(len(l)):
if l[i]%2 ==0:
x=x+1
return x
def g(l,n):
if f(l)==n :
return "Yes"
else:
return "No"
t=int(input())
for _ in range(t):
n=int(input())
l=list(map(int,input().split()))
print(g(l,n))
``` | output | 1 | 28,512 | 22 | 57,025 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3). | instruction | 0 | 28,513 | 22 | 57,026 |
Tags: math
Correct Solution:
```
import sys
input = sys.stdin.readline
n = int(input())
for i in range(n):
m = int(input())
lis = list(map(int,input().split()))
ans_0 = 0
ans_1 = 0
for i in range(len(lis)):
if lis[i]%2 == 0:
ans_0 +=1
else:
ans_1 +=1
if ans_0!= ans_1:
print("No")
else:
print("Yes")
``` | output | 1 | 28,513 | 22 | 57,027 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a multiset (i. e. a set that can contain multiple equal integers) containing 2n integers. Determine if you can split it into exactly n pairs (i. e. each element should be in exactly one pair) so that the sum of the two elements in each pair is odd (i. e. when divided by 2, the remainder is 1).
Input
The input consists of multiple test cases. The first line contains an integer t (1≤ t≤ 100) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer n (1≤ n≤ 100).
The second line of each test case contains 2n integers a_1,a_2,..., a_{2n} (0≤ a_i≤ 100) — the numbers in the set.
Output
For each test case, print "Yes" if it can be split into exactly n pairs so that the sum of the two elements in each pair is odd, and "No" otherwise. You can print each letter in any case.
Example
Input
5
2
2 3 4 5
3
2 3 4 5 5 5
1
2 4
1
2 3
4
1 5 3 2 6 7 3 4
Output
Yes
No
No
Yes
No
Note
In the first test case, a possible way of splitting the set is (2,3), (4,5).
In the second, third and fifth test case, we can prove that there isn't any possible way.
In the fourth test case, a possible way of splitting the set is (2,3).
Submitted Solution:
```
# SHRi GANESHA author: Kunal Verma #
import os
import sys
from bisect import bisect_left, bisect_right
from collections import Counter, defaultdict
from functools import reduce
from io import BytesIO, IOBase
from itertools import combinations
from math import gcd, inf, sqrt, ceil, floor
#sys.setrecursionlimit(2*10**5)
def lcm(a, b):
return (a * b) // gcd(a, b)
'''
mod = 10 ** 9 + 7
fac = [1]
for i in range(1, 2 * 10 ** 5 + 1):
fac.append((fac[-1] * i) % mod)
fac_in = [pow(fac[-1], mod - 2, mod)]
for i in range(2 * 10 ** 5, 0, -1):
fac_in.append((fac_in[-1] * i) % mod)
fac_in.reverse()
def comb(a, b):
if a < b:
return 0
return (fac[a] * fac_in[b] * fac_in[a - b]) % mod
'''
MAXN = 1000004
spf = [0 for i in range(MAXN)]
def sieve():
spf[1] = 1
for i in range(2, MAXN):
spf[i] = i
for i in range(4, MAXN, 2):
spf[i] = 2
for i in range(3, ceil(sqrt(MAXN))):
if (spf[i] == i):
for j in range(i * i, MAXN, i):
if (spf[j] == j):
spf[j] = i
def getFactorization(x):
ret = Counter()
while (x != 1):
ret[spf[x]] += 1
x = x // spf[x]
return ret
def printDivisors(n):
i = 2
z = [1, n]
while i <= sqrt(n):
if (n % i == 0):
if (n / i == i):
z.append(i)
else:
z.append(i)
z.append(n // i)
i = i + 1
return z
def create(n, x, f):
pq = len(bin(n)[2:])
if f == 0:
tt = min
else:
tt = max
dp = [[inf] * n for _ in range(pq)]
dp[0] = x
for i in range(1, pq):
for j in range(n - (1 << i) + 1):
dp[i][j] = tt(dp[i - 1][j], dp[i - 1][j + (1 << (i - 1))])
return dp
def enquiry(l, r, dp, f):
if l > r:
return inf if not f else -inf
if f == 1:
tt = max
else:
tt = min
pq1 = len(bin(r - l + 1)[2:]) - 1
return tt(dp[pq1][l], dp[pq1][r - (1 << pq1) + 1])
def SieveOfEratosthenes(n):
prime = [True for i in range(n + 1)]
p = 2
while (p * p <= n):
if (prime[p] == True):
for i in range(p * p, n + 1, p):
prime[i] = False
p += 1
x = []
for i in range(2, n + 1):
if prime[i]:
x.append(i)
return x
def main():
for _ in range(int(input())):
n=int(input())
a=[int(X) for X in input().split()]
if n>2:
print("No")
else:
od,ev=0,0
for i in a:
if i%2:
od+=1
else:
ev+=1
if od<=2 and ev<=2 and ev<=od:
print("Yes")
else:
print("No")
# Fast IO Region
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)
input = lambda: sys.stdin.readline().rstrip("\r\n")
if __name__ == '__main__':
main()
``` | instruction | 0 | 28,520 | 22 | 57,040 |
No | output | 1 | 28,520 | 22 | 57,041 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,828 | 22 | 57,656 |
Tags: math, number theory
Correct Solution:
```
l,r,x,y=map(int,input().split())
import math
def it(a):
if a<l or a>r:return(False)
return(True)
ans=0
div=set()
for i in range(1,int(y**0.5)+1):
if y%i==0 and (y//i)%x==0 and it(i*x) and it(y//i) and math.gcd(i,(y//i)//x)==1:
c=1 if i==(y//i)//x else 0
ans+=2-c
div.add(y//i)
div.add(i*x)
print(len(div))
``` | output | 1 | 28,828 | 22 | 57,657 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,829 | 22 | 57,658 |
Tags: math, number theory
Correct Solution:
```
from math import sqrt
def p_divs(n):
_a = []
for _i in range(2,int(sqrt(n)) + 10):
while n % _i == 0:
_a.append(_i)
n //= _i
if n > 1:
_a.append(n)
return _a
l, r, x, y = map(int, input().split())
if y % x != 0:
print(0)
else:
m = y // x
divs = p_divs(m)
d = dict()
for p in divs:
if p in d:
d[p] += 1
else:
d[p] = 1
p = []
for k in d:
p.append((k, d[k]))
ans = 0
for i in range(2**len(p)):
a,b = x,x
for k in range(len(p)):
if (i // (2**k)) % 2 == 1:
a *= p[k][0]**p[k][1]
else:
b *= p[k][0]**p[k][1]
if a >= l and a <= r and b >= l and b <= r:
ans += 1
print(ans)
``` | output | 1 | 28,829 | 22 | 57,659 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,830 | 22 | 57,660 |
Tags: math, number theory
Correct Solution:
```
l, r, x, y = map(int, input().split())
f = lambda ka, kb: int(gcd(ka*x,kb*x)==x and l <= ka * x <= r and l <= kb * x <= r)
gcd = lambda a, b: a if b == 0 else gcd(b, a % b)
cnt = 0
k = y // x
d = 1
while d * d <= k:
if k % d == 0:
cnt += f(d, k // d)
if d * d != k: cnt += f(k // d, d)
d += 1
if y % x: cnt = 0
print(cnt)
``` | output | 1 | 28,830 | 22 | 57,661 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,831 | 22 | 57,662 |
Tags: math, number theory
Correct Solution:
```
f=lambda: map(int,input().split())
def nod(a,b):
while True:
if a%b==0:
return b
if b%a==0:
return a
if a>b:
a,b=b,a%b
else:
a,b=b%a,a
l,r,x,y=f()
if y%x!=0:
print(0)
else:
k=0
z = int(y / x)
for c in range(1,int(z**0.5)+1):
if z % c==0:
d = int(z / c)
if l<=c*x and c*x<=r and l<=d*x and d*x<=r and nod(c,d)==1:
k+=1
if c*c!=z: k+=1
print(k)
``` | output | 1 | 28,831 | 22 | 57,663 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,832 | 22 | 57,664 |
Tags: math, number theory
Correct Solution:
```
from sys import stdin, stdout
from math import gcd
input = stdin.buffer.readline
def add(a, b):
if l <= a <= r and l <= b <= r and gcd(a, b) == x:
s.add((a, b))
l, r, x, y = map(int, input().split())
s = set()
a = []
b = []
i = 1
while i * i <= x:
if not x % i:
a.append(i)
a.append(x // i)
add(i, x * y // i)
add(x * y // i, i)
i += 1
i = 1
while i * i <= y:
if not y % i:
b.append(i)
b.append(y // i)
add(i, x * y // i)
add(x * y // i, i)
i += 1
for i in a:
for j in b:
add(i * j, x * y // i // j)
add(x * y // i // j, i * j)
print(len(s))
``` | output | 1 | 28,832 | 22 | 57,665 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,833 | 22 | 57,666 |
Tags: math, number theory
Correct Solution:
```
import math
l, r, gcd, lcm = (int(x) for x in input().split())
if lcm % gcd != 0:
print(0)
quit()
divisors = set()
for div in range(1, int(lcm ** 0.5 + 2)):
if lcm % div == 0:
divisors.add(div)
divisors.add(lcm // div)
prod = lcm * gcd
answers = set()
for div in divisors:
if div < l or div > r:
continue
dop = prod // div
if dop < l or dop > r:
continue
if math.gcd(div, dop) != gcd:
continue
answers.add((div, dop))
answers.add((dop, div))
print(len(answers))
``` | output | 1 | 28,833 | 22 | 57,667 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,834 | 22 | 57,668 |
Tags: math, number theory
Correct Solution:
```
#!/usr/bin/env python3
from sys import exit
from math import sqrt
[l, r, x, y] = map(int, input().strip().split())
if y % x != 0:
print (0)
exit()
y = y // x
l = -((-l) // x) # ceil
r = r // x
pr = []
i = 2
yx = y
mx = sqrt(yx)
while i <= mx:
d = 1
while yx % i == 0:
d *= i
yx //= i
if d > 1:
pr.append(d)
mx = sqrt(yx)
i += 1
if yx > 1:
pr.append(yx)
def count(a, ar):
if len(ar) == 0:
if l <= a <= r and l <= y // a <= r:
return 1
else:
return 0
res = 0
res += count(a, ar[1:])
aa = a * ar[0]
if aa <= r and y // aa >= l:
res += count(aa, ar[1:])
return res
res = count(1, pr)
print (res)
``` | output | 1 | 28,834 | 22 | 57,669 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r. | instruction | 0 | 28,835 | 22 | 57,670 |
Tags: math, number theory
Correct Solution:
```
import os, sys
from io import BytesIO, IOBase
from math import sqrt,ceil,gcd,log2
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 dtb(n):
return bin(n).replace("0b", "")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
def lcm(a,b):
return a*b//gcd(a,b)
l,r,x,y=map(int,input().split())
a=[1]
z=y//x
if y%x:
print(0)
else:
i = 1
t = sqrt(z)
ct=0
while i <= t:
if z % i==0:
if l<=x*i<=r and l<=x*z//i<=r and gcd(i,z//i)==1:
ct+=2
#print(i,z//i,z)
if i==z//i:
ct-=1
i += 1
print(ct)
``` | output | 1 | 28,835 | 22 | 57,671 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
###### ### ####### ####### ## # ##### ### #####
# # # # # # # # # # # # # ###
# # # # # # # # # # # # # ###
###### ######### # # # # # # ######### #
###### ######### # # # # # # ######### #
# # # # # # # # # # #### # # #
# # # # # # # ## # # # # #
###### # # ####### ####### # # ##### # # # #
from __future__ import print_function # for PyPy2
# from itertools import permutations
# from functools import cmp_to_key # for adding custom comparator
# from fractions import Fraction
from collections import *
from sys import stdin
# from bisect import *
from heapq import *
from math import *
g = lambda : stdin.readline().strip()
gl = lambda : g().split()
gil = lambda : [int(var) for var in gl()]
gfl = lambda : [float(var) for var in gl()]
gcl = lambda : list(g())
gbs = lambda : [int(var) for var in g()]
rr = lambda x : reversed(range(x))
mod = int(1e9)+7
inf = float("inf")
l, r, gc, lc = gil()
if lc%gc :print(0);exit()
k = lc//gc # k1*k2
ans = 0
for k1 in range(1, int((sqrt(k)))+1):
if k%k1 : continue
k2 = k//k1
if gcd(k1, k2) > 1 : continue
if l <= gc*k1 <= r and l <= gc*k2 <= r:
ans += 1 if k1 == k2 else 2
print(ans)
``` | instruction | 0 | 28,836 | 22 | 57,672 |
Yes | output | 1 | 28,836 | 22 | 57,673 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
from fractions import gcd
l,r,x,y= map(int,input().split())
coun=0
li=[1,y]
i=2
while i*i <=y:
if y%i==0 and i*i!=y:
li.append(i); li.append(y/i)
elif i*i==y: li.append(i)
i+=1
for j in li:
if j>=l and j<=r and j%x==0:
a= (x*y)//j
if a>=l and a<=r and gcd(a,j)==x:
coun+=1
#print(a,j)
print(coun)
``` | instruction | 0 | 28,837 | 22 | 57,674 |
Yes | output | 1 | 28,837 | 22 | 57,675 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
import sys
from math import sqrt,ceil,gcd
def sieve(N):
b = [True]*(N+1)
b[0] = False
b[1] = False
lim = ceil(sqrt(N))
i = 2
while i <= lim:
if b[i]:
for n in range(i**2,N+1,i):
b[n] = False
i+=1
return [i for i,b in enumerate(b) if b]
P = sieve(10**5)
def factor(n,P):
"""Given prime list, factorize n"""
if n in P: return [n]
f = []
for p in P:
while n%p == 0:
n//=p
f.append(p)
if n in P:
f.append(n)
return f
if n != 1:
f.append(n)
return f
def divisors(n):
F = factor(n,P)
D = {1}
for f in F:
D |= {f*d for d in D}
return D
l,r,x,y = map(int,input().split())
a = x
if y%a != 0:
print(0)
sys.exit()
x//=a
y//=a
cnt = 0
for d in divisors(y):
n = d*a
m = y//d*a
if l<=n<=r and l<=m<=r and gcd(d,y//d) == 1:
cnt += 1
print(cnt)
``` | instruction | 0 | 28,838 | 22 | 57,676 |
Yes | output | 1 | 28,838 | 22 | 57,677 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
import math
from fractions import gcd
l,r,x,y=list(map(int,input().strip().split()))
b=math.ceil(l/x)
e=r//x
if (y%x!=0):
print (0)
exit(0)
y=y/x
if (y==1 and b==1 and e>=1):
print (1)
exit(0)
ans=0
for i in range(1,math.ceil(math.sqrt(y))):
if (y%i)==0:
d=y//i
if (gcd(i,d)==1 and i>=b and i<=e and d>=b and d<=e):
if (i==d):
ans=ans+1
else:
ans=ans+2
print (ans)
``` | instruction | 0 | 28,839 | 22 | 57,678 |
Yes | output | 1 | 28,839 | 22 | 57,679 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
import math
l,r,x,y = map(int,input().split())
if y%x!=0:
print(0)
quit()
aa=y//x
ans=0
l=l/x;r=r/x;
for i in range(1,int(math.sqrt(y/x))+1,1):
if aa%i==0:
a,b = i , (aa//i)
if l<=a<=r and l<=b<=r:
if math.gcd(a,b)==1:
ans+=2
if a==b:
ans-=1
print(ans)
# print(2*len(lis),lis)
``` | instruction | 0 | 28,840 | 22 | 57,680 |
No | output | 1 | 28,840 | 22 | 57,681 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
import math
from fractions import gcd
x,y,l,r=list(map(int,input().strip().split()))
b=math.ceil(l/x)
e=r//x
if (y%x!=0):
print (0)
exit(0)
y=y/x
ans=0
for i in range(1,int(math.sqrt(y))+1):
if (y%i)==0:
d=y//i
if (gcd(i,d)==1 and i>=b and i<=e and d>=b and d<=e):
if (i==d):
ans=ans+1
else:
ans=ans+2
print (ans)
``` | instruction | 0 | 28,841 | 22 | 57,682 |
No | output | 1 | 28,841 | 22 | 57,683 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
import math
def pr(n) :
a=[]
# Note that this loop runs till square root
i = 1
while i <= math.sqrt(n):
if (n % i == 0) :
# If divisors are equal, print only one
if (n / i == i) :
a.append(i)
else :
# Otherwise print both
a.extend([i , n//i])
i = i + 1
return a
def gcd(a,b):
if a%b==0:
return b
else:
return gcd(b,a%b)
def fu(a,b,c):
d=pr(c)
d.sort()
e=[]
for i in range(len(d)):
if d[i]<=a:
e.append(d[i])
else:
break
d=0
for i in range(len(e)):
for j in range(len(e)):
if gcd(e[i],e[j])==b and (e[i]*e[j])//b==c:
d+=1
return d
l,r,x,y=list(map(int,input().split()))
if [l,r,x,y]==[39443,809059020,19716,777638472]:
print(12)
else:
print(fu(r,x,y)-fu(l-1,x,y))
``` | instruction | 0 | 28,842 | 22 | 57,684 |
No | output | 1 | 28,842 | 22 | 57,685 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on Informatics class Nastya learned about GCD and LCM (see links below). Nastya is very intelligent, so she solved all the tasks momentarily and now suggests you to solve one of them as well.
We define a pair of integers (a, b) good, if GCD(a, b) = x and LCM(a, b) = y, where GCD(a, b) denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of a and b, and LCM(a, b) denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of a and b.
You are given two integers x and y. You are to find the number of good pairs of integers (a, b) such that l ≤ a, b ≤ r. Note that pairs (a, b) and (b, a) are considered different if a ≠ b.
Input
The only line contains four integers l, r, x, y (1 ≤ l ≤ r ≤ 109, 1 ≤ x ≤ y ≤ 109).
Output
In the only line print the only integer — the answer for the problem.
Examples
Input
1 2 1 2
Output
2
Input
1 12 1 12
Output
4
Input
50 100 3 30
Output
0
Note
In the first example there are two suitable good pairs of integers (a, b): (1, 2) and (2, 1).
In the second example there are four suitable good pairs of integers (a, b): (1, 12), (12, 1), (3, 4) and (4, 3).
In the third example there are good pairs of integers, for example, (3, 30), but none of them fits the condition l ≤ a, b ≤ r.
Submitted Solution:
```
s=list(map(int, input().split()))
l=s[0]
r=s[1]
x=s[2]
y=s[3]
def findgcd(a,b):
M=max(a,b)
m=min(a,b)
while(m!=0):
t=M%m
M=m
m=t
return M
count=0
if(x!=y):
if(True):
prod=x*y
for i in range(1,int(prod**0.5)+1):
k=prod/i
if(findgcd(k,i)==x):
if(i==k):
count+=1
else:
count+=2
if(x==y):
count=1
print(count)
``` | instruction | 0 | 28,843 | 22 | 57,686 |
No | output | 1 | 28,843 | 22 | 57,687 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,924 | 22 | 57,848 |
"Correct Solution:
```
N = int(input())
if N % 2 != 0:
N = N * 2
print(N)
``` | output | 1 | 28,924 | 22 | 57,849 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,925 | 22 | 57,850 |
"Correct Solution:
```
i=int(input())
print(i if i%2==0 else i*2)
``` | output | 1 | 28,925 | 22 | 57,851 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,926 | 22 | 57,852 |
"Correct Solution:
```
a = input()
a = int(a)
if a%2 ==0:print(a)
else:print(2*a)
``` | output | 1 | 28,926 | 22 | 57,853 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,927 | 22 | 57,854 |
"Correct Solution:
```
n = int(input())
print(n if (n % 2 == 0) else (n * 2))
``` | output | 1 | 28,927 | 22 | 57,855 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,928 | 22 | 57,856 |
"Correct Solution:
```
n = int(input())
if n%2:
n*=2
print(n)
``` | output | 1 | 28,928 | 22 | 57,857 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,929 | 22 | 57,858 |
"Correct Solution:
```
import math
n = int(input())
print(2 * n // math.gcd(2, n))
``` | output | 1 | 28,929 | 22 | 57,859 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,930 | 22 | 57,860 |
"Correct Solution:
```
n = int(input())
ans = n
if n%2:
ans *= 2
print(ans)
``` | output | 1 | 28,930 | 22 | 57,861 |
Provide a correct Python 3 solution for this coding contest problem.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998 | instruction | 0 | 28,931 | 22 | 57,862 |
"Correct Solution:
```
n=int(input())
if n%2:
ans=2*n
else:
ans=n
print(ans)
``` | output | 1 | 28,931 | 22 | 57,863 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
import math
N = int(input())
print((2*N)//math.gcd(2,N))
``` | instruction | 0 | 28,932 | 22 | 57,864 |
Yes | output | 1 | 28,932 | 22 | 57,865 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
n=int(input())
print(2*n) if n%2 != 0 else print(n)
``` | instruction | 0 | 28,933 | 22 | 57,866 |
Yes | output | 1 | 28,933 | 22 | 57,867 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
s = int(input())
print(s if s%2 == 0 else s*2)
``` | instruction | 0 | 28,934 | 22 | 57,868 |
Yes | output | 1 | 28,934 | 22 | 57,869 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
a=int(input())
if a%2==0:
print(a)
else:
print(2*a)
``` | instruction | 0 | 28,935 | 22 | 57,870 |
Yes | output | 1 | 28,935 | 22 | 57,871 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
import numpy as np
N = int(input())
list_ai = list(map(int,input().split()))
a1 = np.array(list_ai)-np.arange(1,N+1)
median1 = int(np.floor(np.median(a1)))
ans = np.sum(np.abs(a1-median1))
print(ans)
``` | instruction | 0 | 28,936 | 22 | 57,872 |
No | output | 1 | 28,936 | 22 | 57,873 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
n = int(input())
if n%2 == 0:
print(n*2)
else:
print(n)
``` | instruction | 0 | 28,937 | 22 | 57,874 |
No | output | 1 | 28,937 | 22 | 57,875 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
N = input()
print(N*2)
``` | instruction | 0 | 28,938 | 22 | 57,876 |
No | output | 1 | 28,938 | 22 | 57,877 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given a positive integer N. Find the minimum positive integer divisible by both 2 and N.
Constraints
* 1 \leq N \leq 10^9
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N
Output
Print the minimum positive integer divisible by both 2 and N.
Examples
Input
3
Output
6
Input
10
Output
10
Input
999999999
Output
1999999998
Submitted Solution:
```
L = 0
I = 1
N = int(input())
if N < 1 or N > 10:
print('invalido!')
if (N >= 1 and N <= 10):
while L == 0:
if (I % 2 == 0 and I % N == 0):
print(I)
L = 1
I +=I
``` | instruction | 0 | 28,939 | 22 | 57,878 |
No | output | 1 | 28,939 | 22 | 57,879 |
Provide a correct Python 3 solution for this coding contest problem.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6 | instruction | 0 | 29,030 | 22 | 58,060 |
"Correct Solution:
```
# AOJ 1109: Fermat's Last Theorem
# Python3 2018.7.14 bal4u
from bisect import bisect_left
tbl = [i**3 for i in range(1111)]
while True:
z = int(input())
if z == 0: break
ma = 0
for x in range(z-1, 0, -1):
y = bisect_left(tbl, tbl[z]-tbl[x])-1
if tbl[y] > tbl[x]: break
ma = max(ma, tbl[x]+tbl[y])
print(tbl[z]-ma)
``` | output | 1 | 29,030 | 22 | 58,061 |
Provide a correct Python 3 solution for this coding contest problem.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6 | instruction | 0 | 29,031 | 22 | 58,062 |
"Correct Solution:
```
a=1/3
while 1:
z=int(input())
if z==0:break
m=0
zz=z*z*z
for x in range(1,int(z/pow(2,a))+1):
xx=x*x*x
y=int(pow(zz-xx,a))
yy=y*y*y
m=max(m,yy+xx)
print(zz-m)
``` | output | 1 | 29,031 | 22 | 58,063 |
Provide a correct Python 3 solution for this coding contest problem.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6 | instruction | 0 | 29,032 | 22 | 58,064 |
"Correct Solution:
```
while 1:
z=int(input())**3
if z==0:break
m,x=0,0
while 1:
x+=1
xx=x**3
y=int((z-xx)**(1/3))
if y<x:break
m=max(m,y**3+xx)
print(z-m)
``` | output | 1 | 29,032 | 22 | 58,065 |
Provide a correct Python 3 solution for this coding contest problem.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6 | instruction | 0 | 29,033 | 22 | 58,066 |
"Correct Solution:
```
a=1/3
while 1:
z=int(input())
if z==0:break
m,zz=0,z*z*z
for x in range(1,int(z/pow(2,a))+1):
xx=x*x*x
y=int(pow(zz-xx,a))
yy=y*y*y
m=max(m,yy+xx)
print(zz-m)
``` | output | 1 | 29,033 | 22 | 58,067 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6
Submitted Solution:
```
from itertools import combinations_with_replacement as C
while True:
z = int(input())
if z == 0:
break
zzz = pow(z, 3)
candidates = []
combinations = list(C(list(range(1, z+1)), 2))
for xy in combinations:
x, y = xy[0], xy[1]
tmp = pow(x, 3)+pow(y, 3)
if zzz >= tmp:
candidates.append(tmp)
print(zzz-max(candidates))
``` | instruction | 0 | 29,034 | 22 | 58,068 |
No | output | 1 | 29,034 | 22 | 58,069 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6
Submitted Solution:
```
from itertools import combinations_with_replacement as C
while True:
z = int(input())
if z == 0:
break
zzz = pow(z, 3)
candidates = []
combinations = list(C(list(range(1, z+1)), 2))
if len(combinations) >= 100:
conbinations = combinations[100:]
for xy in combinations:
x, y = xy[0], xy[1]
tmp = pow(x, 3)+pow(y, 3)
if zzz >= tmp:
candidates.append(tmp)
print(zzz-max(candidates))
``` | instruction | 0 | 29,035 | 22 | 58,070 |
No | output | 1 | 29,035 | 22 | 58,071 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
In the 17th century, Fermat wrote that he proved for any integer $n \geq 3$, there exist no positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. However he never disclosed the proof. Later, this claim was named Fermat's Last Theorem or Fermat's Conjecture.
If Fermat's Last Theorem holds in case of $n$, then it also holds in case of any multiple of $n$. Thus it suffices to prove cases where $n$ is a prime number and the special case $n$ = 4.
A proof for the case $n$ = 4 was found in Fermat's own memorandum. The case $n$ = 3 was proved by Euler in the 18th century. After that, many mathematicians attacked Fermat's Last Theorem. Some of them proved some part of the theorem, which was a partial success. Many others obtained nothing. It was a long history. Finally, Wiles proved Fermat's Last Theorem in 1994.
Fermat's Last Theorem implies that for any integers $n \geq 3$ and $z > 1$, it always holds that
$z^n > $ max { $x^n + y^n | x > 0, y > 0, x^n + y^n \leq z^n$ }.
Your mission is to write a program that verifies this in the case $n$ = 3 for a given $z$. Your program should read in integer numbers greater than 1, and, corresponding to each input $z$, it should output the following:
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
Input
The input is a sequence of lines each containing one positive integer number followed by a line containing a zero. You may assume that all of the input integers are greater than 1 and less than 1111.
Output
The output should consist of lines each containing a single integer number. Each output integer should be
$z^3 - $ max { $x^3 + y^3 | x > 0, y > 0, x^3 + y^3 \leq z^3$ }.
for the corresponding input integer z. No other characters should appear in any output line.
Example
Input
6
4
2
0
Output
27
10
6
Submitted Solution:
```
from itertools import combinations_with_replacement as C
while True:
z = int(input())
if z == 0:
break
zzz = pow(z, 3)
candidates = []
combinations = list(C(list(range(1, z+1)), 2))
if z >= 100:
mid = len(combinations) // 2
combinations = combinations[mid:]
for xy in combinations:
x, y = xy[0], xy[1]
tmp = pow(x, 3)+pow(y, 3)
if zzz >= tmp:
candidates.append(tmp)
print(zzz-max(candidates))
``` | instruction | 0 | 29,036 | 22 | 58,072 |
No | output | 1 | 29,036 | 22 | 58,073 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a prime number p, n integers a_1, a_2, …, a_n, and an integer k.
Find the number of pairs of indexes (i, j) (1 ≤ i < j ≤ n) for which (a_i + a_j)(a_i^2 + a_j^2) ≡ k mod p.
Input
The first line contains integers n, p, k (2 ≤ n ≤ 3 ⋅ 10^5, 2 ≤ p ≤ 10^9, 0 ≤ k ≤ p-1). p is guaranteed to be prime.
The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i ≤ p-1). It is guaranteed that all elements are different.
Output
Output a single integer — answer to the problem.
Examples
Input
3 3 0
0 1 2
Output
1
Input
6 7 2
1 2 3 4 5 6
Output
3
Note
In the first example:
(0+1)(0^2 + 1^2) = 1 ≡ 1 mod 3.
(0+2)(0^2 + 2^2) = 8 ≡ 2 mod 3.
(1+2)(1^2 + 2^2) = 15 ≡ 0 mod 3.
So only 1 pair satisfies the condition.
In the second example, there are 3 such pairs: (1, 5), (2, 3), (4, 6). | instruction | 0 | 29,199 | 22 | 58,398 |
Tags: math, matrices, number theory, two pointers
Correct Solution:
```
#572_E
ln = [int(i) for i in input().split(" ")]
n = ln[0]
p = ln[1]
k = ln[2]
a = [int(i) for i in input().split(" ")]
cts = {}
ctsc = {}
ans = []
nm = 0
for i in range(0, len(a)):
an = ((a[i] ** 4) - (k * a[i])) % p
if an in ctsc:
nm += ctsc[an]
ctsc[an] += 1
else:
ctsc[an] = 1
print(nm)
``` | output | 1 | 29,199 | 22 | 58,399 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given a prime number p, n integers a_1, a_2, …, a_n, and an integer k.
Find the number of pairs of indexes (i, j) (1 ≤ i < j ≤ n) for which (a_i + a_j)(a_i^2 + a_j^2) ≡ k mod p.
Input
The first line contains integers n, p, k (2 ≤ n ≤ 3 ⋅ 10^5, 2 ≤ p ≤ 10^9, 0 ≤ k ≤ p-1). p is guaranteed to be prime.
The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i ≤ p-1). It is guaranteed that all elements are different.
Output
Output a single integer — answer to the problem.
Examples
Input
3 3 0
0 1 2
Output
1
Input
6 7 2
1 2 3 4 5 6
Output
3
Note
In the first example:
(0+1)(0^2 + 1^2) = 1 ≡ 1 mod 3.
(0+2)(0^2 + 2^2) = 8 ≡ 2 mod 3.
(1+2)(1^2 + 2^2) = 15 ≡ 0 mod 3.
So only 1 pair satisfies the condition.
In the second example, there are 3 such pairs: (1, 5), (2, 3), (4, 6). | instruction | 0 | 29,200 | 22 | 58,400 |
Tags: math, matrices, number theory, two pointers
Correct Solution:
```
import sys
from collections import Counter
n,p,k=[int(x) for x in sys.stdin.readline().strip().split()]
arr=[int(x) for x in sys.stdin.readline().strip().split()]
brr = [int((x**4 - k*x + 100000000*p)%p) for x in arr]
a=Counter(brr)
print(sum([int((int(a[i])*(int(a[i])-1))/2) for i in set(brr)]))
``` | output | 1 | 29,200 | 22 | 58,401 |
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