message stringlengths 2 57.2k | message_type stringclasses 2 values | message_id int64 0 1 | conversation_id int64 61 108k | cluster float64 22 22 | __index_level_0__ int64 122 217k |
|---|---|---|---|---|---|
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute 1 + 3 using the calculator, he gets 2 instead of 4. But when he tries computing 1 + 4, he gets the correct answer, 5. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders!
So, when he tries to compute the sum a + b using the calculator, he instead gets the xorsum a ⊕ b (read the definition by the link: <https://en.wikipedia.org/wiki/Exclusive_or>).
As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers l and r, how many pairs of integers (a, b) satisfy the following conditions: $$$a + b = a ⊕ b l ≤ a ≤ r l ≤ b ≤ r$$$
However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.
Input
The first line contains a single integer t (1 ≤ t ≤ 100) — the number of testcases.
Then, t lines follow, each containing two space-separated integers l and r (0 ≤ l ≤ r ≤ 10^9).
Output
Print t integers, the i-th integer should be the answer to the i-th testcase.
Example
Input
3
1 4
323 323
1 1000000
Output
8
0
3439863766
Note
a ⊕ b denotes the bitwise XOR of a and b.
For the first testcase, the pairs are: (1, 2), (1, 4), (2, 1), (2, 4), (3, 4), (4, 1), (4, 2), and (4, 3).
Submitted Solution:
```
from sys import stderr
max_bit = 40
comb = [[0] * max_bit for i in range(max_bit)]
for i in range(max_bit):
comb[i][0] = 1
comb[i][i] = 1
for f in range(1, i):
comb[i][f] = (comb[i - 1][f] + comb[i - 1][f - 1])
def cal_num_bit(upper_bound):
ans = [0] * max_bit
filled_bit = 0
cur_num = 0
for i in range(max_bit - 1, -1, -1):
next_num = cur_num + (1 << i)
if next_num <= upper_bound:
for f in range(filled_bit, max_bit):
ans[f] += comb[i][f]
filled_bit += 1
cur_num = next_num
return ans
def solve(a, b):
bin_a, bin_b = [], []
for i in range(max_bit):
bin_a.append(a & 1)
bin_b.append(b & 1)
a >>= 1
b >>= 1
while len(bin_a) > 0 and bin_a[-1] == 0 and bin_b[-1] == 0:
bin_a.pop()
bin_b.pop()
if len(bin_a) == 0:
return 1
dp = [[[-1, -1], [-1, -1]] for i in range(max_bit)]
def cal_dp(pos, eq_upper, eq_lower):
if pos == -1:
return 1
if dp[pos][int(eq_upper)][int(eq_lower)] != -1:
return dp[pos][int(eq_upper)][int(eq_lower)]
upper_bit = bin_b[pos] if eq_upper else 1
lower_bit = bin_a[pos] if eq_lower else 0
dp[pos][int(eq_upper)][int(eq_lower)] = 0
for bigger in range(0, upper_bit + 1):
for smaller in range(lower_bit, 2):
if bigger != 0 and smaller != 0:
continue
new_eq_upper = bigger == upper_bit if eq_upper else False
new_eq_lower = smaller == lower_bit if eq_lower else False
dp[pos][int(eq_upper)][int(eq_lower)] += cal_dp(pos - 1, new_eq_upper, new_eq_lower)
return dp[pos][int(eq_upper)][int(eq_lower)]
ans = 0
has_diff = False
for i in range(len(bin_a) - 1, -1, -1):
if bin_a[i] == 1:
break
ans += cal_dp(i - 1, not has_diff, True)
if bin_a[i] != bin_b[i]:
has_diff = True
return ans * 2
for test in range(int(input())):
print(solve(*map(int, input().split())))
``` | instruction | 0 | 76,835 | 22 | 153,670 |
No | output | 1 | 76,835 | 22 | 153,671 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute 1 + 3 using the calculator, he gets 2 instead of 4. But when he tries computing 1 + 4, he gets the correct answer, 5. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders!
So, when he tries to compute the sum a + b using the calculator, he instead gets the xorsum a ⊕ b (read the definition by the link: <https://en.wikipedia.org/wiki/Exclusive_or>).
As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers l and r, how many pairs of integers (a, b) satisfy the following conditions: $$$a + b = a ⊕ b l ≤ a ≤ r l ≤ b ≤ r$$$
However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.
Input
The first line contains a single integer t (1 ≤ t ≤ 100) — the number of testcases.
Then, t lines follow, each containing two space-separated integers l and r (0 ≤ l ≤ r ≤ 10^9).
Output
Print t integers, the i-th integer should be the answer to the i-th testcase.
Example
Input
3
1 4
323 323
1 1000000
Output
8
0
3439863766
Note
a ⊕ b denotes the bitwise XOR of a and b.
For the first testcase, the pairs are: (1, 2), (1, 4), (2, 1), (2, 4), (3, 4), (4, 1), (4, 2), and (4, 3).
Submitted Solution:
```
print("8\n0\n3439863766")
``` | instruction | 0 | 76,836 | 22 | 153,672 |
No | output | 1 | 76,836 | 22 | 153,673 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute 1 + 3 using the calculator, he gets 2 instead of 4. But when he tries computing 1 + 4, he gets the correct answer, 5. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders!
So, when he tries to compute the sum a + b using the calculator, he instead gets the xorsum a ⊕ b (read the definition by the link: <https://en.wikipedia.org/wiki/Exclusive_or>).
As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers l and r, how many pairs of integers (a, b) satisfy the following conditions: $$$a + b = a ⊕ b l ≤ a ≤ r l ≤ b ≤ r$$$
However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.
Input
The first line contains a single integer t (1 ≤ t ≤ 100) — the number of testcases.
Then, t lines follow, each containing two space-separated integers l and r (0 ≤ l ≤ r ≤ 10^9).
Output
Print t integers, the i-th integer should be the answer to the i-th testcase.
Example
Input
3
1 4
323 323
1 1000000
Output
8
0
3439863766
Note
a ⊕ b denotes the bitwise XOR of a and b.
For the first testcase, the pairs are: (1, 2), (1, 4), (2, 1), (2, 4), (3, 4), (4, 1), (4, 2), and (4, 3).
Submitted Solution:
```
n = int(input())
for i in range(n):
l, r = [int(i) for i in input().split()]
k = 0
if l == 0:
l = 1
for j in range(l + 1, r + 1):
k += (2 ** bin(j).split("0b")[1].count("0") - l) * 2
print(k)
``` | instruction | 0 | 76,837 | 22 | 153,674 |
No | output | 1 | 76,837 | 22 | 153,675 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute 1 + 3 using the calculator, he gets 2 instead of 4. But when he tries computing 1 + 4, he gets the correct answer, 5. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders!
So, when he tries to compute the sum a + b using the calculator, he instead gets the xorsum a ⊕ b (read the definition by the link: <https://en.wikipedia.org/wiki/Exclusive_or>).
As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers l and r, how many pairs of integers (a, b) satisfy the following conditions: $$$a + b = a ⊕ b l ≤ a ≤ r l ≤ b ≤ r$$$
However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.
Input
The first line contains a single integer t (1 ≤ t ≤ 100) — the number of testcases.
Then, t lines follow, each containing two space-separated integers l and r (0 ≤ l ≤ r ≤ 10^9).
Output
Print t integers, the i-th integer should be the answer to the i-th testcase.
Example
Input
3
1 4
323 323
1 1000000
Output
8
0
3439863766
Note
a ⊕ b denotes the bitwise XOR of a and b.
For the first testcase, the pairs are: (1, 2), (1, 4), (2, 1), (2, 4), (3, 4), (4, 1), (4, 2), and (4, 3).
Submitted Solution:
```
from math import *
def t1(p,h):
if p==0:
return h
if h>2**(p-1):
return 2*3**(p-1)+t1(p-1,h-2**(p-1))
else:
return 2*t1(p-1,h)
def t(a,b):
if a==0 or b==0:
return 0
if a>b:
return t(b,a)
m=1
e=0
while 2*m<=a:
m*=2
e+=1
r=b//m
return r*t1(e,m)+t1(e,b%m)+(r+1)//2*t1(e,a-m)
a=int(input())
for i in range(a):
x,y=[int(s) for s in input().split()]
print(t(y+1,y+1)-2*t(x,y+1)+t(x,x))
``` | instruction | 0 | 76,838 | 22 | 153,676 |
No | output | 1 | 76,838 | 22 | 153,677 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,357 | 22 | 154,714 |
"Correct Solution:
```
n, k = map(int, input().split())
mod = 10**9+7
dp = [0]*(k+1)
ans = 0
for i in range(1, k+1)[::-1]:
res = pow(k//i, n, mod)
for j in range(i*2, k+1, i):
res-=dp[j]
dp[i] = res%mod
ans+=i*res
ans%=mod
print(ans)
``` | output | 1 | 77,357 | 22 | 154,715 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,358 | 22 | 154,716 |
"Correct Solution:
```
mod = 10 ** 9 + 7
N, K = map(int, input().split())
ans = 0
memo = [0] * (K + 1)
for i in range(K, 0, -1):
x = K // i
cnt = pow(x, N, mod)
a = i
while a <= K:
cnt -= memo[a]
a += i
memo[i] = cnt
ans += i * cnt
ans %= mod
print(ans)
``` | output | 1 | 77,358 | 22 | 154,717 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,359 | 22 | 154,718 |
"Correct Solution:
```
n,k=map(int,input().split())
mod=pow(10,9)+7
ans=[0]*(k+1)
# gcdがkiとなる数列。すべてがkiの倍数でかつ少なくとも一つkiを含む
for ki in range(k,0,-1):
a=k//ki
ans[ki]=pow(a,n,mod)
i=2
while i*ki<=k:
ans[ki]-=ans[i*ki]
i+=1
b=0
for ki in range(1,k+1):
b+=(ans[ki]*ki)%mod
b%=mod
print(b)
``` | output | 1 | 77,359 | 22 | 154,719 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,360 | 22 | 154,720 |
"Correct Solution:
```
n,k=map(int,input().split())
mod=10**9+7
ans=0
d=[0]*(k+1)
for i in range(k,0,-1):
c=k//i
t=pow(c,n,mod)
t+=(d[i]//mod+1)*mod-d[i]
t%=mod
ans+=t*i
ans%=mod
for j in range(1,int(i**.5)+1):
if i%j==0:
d[j]+=t
if j!=i//j:
d[i//j]+=t
print(ans)
``` | output | 1 | 77,360 | 22 | 154,721 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,361 | 22 | 154,722 |
"Correct Solution:
```
from math import gcd
N,K = map(int,input().split())
MOD = 10**9+7
dp = [1] * (K+1)
for n in range(K//2,0,-1):
p = pow(K//n,N,MOD)
for m in range(2*n,K+1,n):
p -= dp[m]
dp[n] = p%MOD
ans = 0
for i,n in enumerate(dp):
ans += i*n
ans %= MOD
print(ans)
``` | output | 1 | 77,361 | 22 | 154,723 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,362 | 22 | 154,724 |
"Correct Solution:
```
n,k=map(int,input().split())
mod=10**9+7
ans=0
A=[0]*k
for i in range(k,0,-1):
a=0
A[i-1]=pow((k//i),n,mod)
m=i*2
while m<=k:
A[i-1]=(A[i-1]-A[m-1])%mod
m=m+i
ans=(ans+i*A[i-1])%mod
print(ans%mod)
``` | output | 1 | 77,362 | 22 | 154,725 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,363 | 22 | 154,726 |
"Correct Solution:
```
N, K = [int(_) for _ in input().split()]
mod = 10**9 + 7
A = [0] * (K + 1)
for i in range(K, 0, -1):
A[i] = pow(K // i, N, mod)
for j in range(2, K // i + 1):
A[i] -= A[i * j]
A[i] %= mod
print(sum(i * a for i, a in enumerate(A)) % mod)
``` | output | 1 | 77,363 | 22 | 154,727 |
Provide a correct Python 3 solution for this coding contest problem.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979 | instruction | 0 | 77,364 | 22 | 154,728 |
"Correct Solution:
```
N,K=map(int,input().split())
m=10**9+7
d=[0]*K
for k in range(K,0,-1):
d[k-1]=pow(K//k,N,m)
for l in range(2*k,K+1,k):
d[k-1]=(d[k-1]-d[l-1])%m
print(sum([(k+1)*d[k]%m for k in range(K)])%m)
``` | output | 1 | 77,364 | 22 | 154,729 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
n,k=map(int,input().split())
mod=10**9+7
count=[0]*(k+1)
def getnum(m):
ret = pow(k//m,n,mod)
mul=2
while m*mul<=k:
ret-=count[m*mul]
mul+=1
return ret%mod
ans=0
for i in range(1,k+1)[::-1]:
g=getnum(i)
count[i]=g
ans+=g*i
ans%=mod
print(ans)
``` | instruction | 0 | 77,365 | 22 | 154,730 |
Yes | output | 1 | 77,365 | 22 | 154,731 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
n, k = map(int, input().split())
p = 10 ** 9 + 7
cnt = [0] * (k + 1)
for i in range(k, 0, -1):
cnt[i] = pow((k // i), n, p)
for j in range(i * 2, k + 1, i):
cnt[i] -= cnt[j]
print(sum((i * cnt[i] for i in range(1, k + 1))) % p)
``` | instruction | 0 | 77,366 | 22 | 154,732 |
Yes | output | 1 | 77,366 | 22 | 154,733 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
n,k=map(int,input().split())
mod=10**9+7
lst=[0]*(k+1)
ans=0
for i in range(k,0,-1):
lst[i]+=pow(k//i,n,mod)
if k//i==1:
continue
else:
for j in range(2,k//i+1):
lst[i]-=lst[i*j]
for i in range(1,k+1):
ans+=(i*lst[i])%mod
ans%=mod
print(ans)
``` | instruction | 0 | 77,367 | 22 | 154,734 |
Yes | output | 1 | 77,367 | 22 | 154,735 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
N, K = map(int, input().split())
MOD = 10**9 + 7
cnt = [0] * (K + 1)
def calc(x):
M = K // x
c = pow(M, N, MOD)
for i in range(x + x, K + 1, x):
c -= cnt[i]
cnt[x] = c
return c * x
ans = 0
for x in range(1, K + 1)[::-1]:
ans = (ans + calc(x)) % MOD
print(ans)
``` | instruction | 0 | 77,368 | 22 | 154,736 |
Yes | output | 1 | 77,368 | 22 | 154,737 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
N, K = map(int, input().split())
p = int(10e9+7)
res = 0
c = 0
d = [0]*K
for i in range(K, 0, -1):
n = ((K//i)**N)
j = 2
x = i*j
while x <= K:
n -= d[x-1]
j += 1
x = i*j
d[i-1] = n
res += n*i
# res += xn*i
res %= p
print(res)
``` | instruction | 0 | 77,369 | 22 | 154,738 |
No | output | 1 | 77,369 | 22 | 154,739 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
def divisor_enumetarion(n):
divisors = []
for i in range(1, int(n**0.5)+1):
if n % i == 0:
divisors.append(i)
if i != n // i:
divisors.append(n//i)
divisors.remove(n)
return divisors
n,k=map(int,input().split())
dp=[0]*k
ans=0
for i in range(k,0,-1):
toori=(k//i)**n
dp[i-1]+=toori
x=divisor_enumetarion(i)
for j in x:
dp[j-1]-=dp[i-1]
ans+=dp[i-1]*i
# print(dp)
# print(i,dp[i-1])
print(ans%(10**9+7))
``` | instruction | 0 | 77,370 | 22 | 154,740 |
No | output | 1 | 77,370 | 22 | 154,741 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
def gcd(x, y):
while (y):
x, y = y, x % y
return x
MOD = int(10e9 + 7)
n, k = map(int, input().split())
mem = {}
def dp(at, cur):
if at == n:
return cur
if (at,cur) in mem:
return mem[(at,cur)]
sm = 0
for i in range(1, k+1):
sm += dp(at+1,gcd(cur, i))
mem[(at,cur)] = sm % MOD
return sm % MOD
sm = 0
for i in range(1, k+1):
sm += dp(1, i)
print(sm % MOD)
``` | instruction | 0 | 77,371 | 22 | 154,742 |
No | output | 1 | 77,371 | 22 | 154,743 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Consider sequences \\{A_1,...,A_N\\} of length N consisting of integers between 1 and K (inclusive).
There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.
Since this sum can be enormous, print the value modulo (10^9+7).
Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.
Constraints
* 2 \leq N \leq 10^5
* 1 \leq K \leq 10^5
* All values in input are integers.
Input
Input is given from Standard Input in the following format:
N K
Output
Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).
Examples
Input
3 2
Output
9
Input
3 200
Output
10813692
Input
100000 100000
Output
742202979
Submitted Solution:
```
#!/usr/bin/env python3
import sys
MOD = 1000000007 # type: int
def solve(N: int, K: int):
dictionary = {} # key k番目には、kを最小公倍数として持つような組合せの数が保存されている。
for k in range(K, 0, -1):
dictionary[k] = (int(K / k) ** N)
multiplier = 2
while k * multiplier <= K:
dictionary[k] -= dictionary[k * multiplier]
multiplier += 1
result = sum([k * v for k, v in dictionary.items()])
print(result % (10**9+7))
return
# Generated by 1.1.6 https://github.com/kyuridenamida/atcoder-tools (tips: You use the default template now. You can remove this line by using your custom template)
def main():
def iterate_tokens():
for line in sys.stdin:
for word in line.split():
yield word
tokens = iterate_tokens()
N = int(next(tokens)) # type: int
K = int(next(tokens)) # type: int
solve(N, K)
if __name__ == '__main__':
main()
``` | instruction | 0 | 77,372 | 22 | 154,744 |
No | output | 1 | 77,372 | 22 | 154,745 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,476 | 22 | 154,952 |
"Correct Solution:
```
def main():
while True:
try:
a,b=(int(x) for x in input().split())
except:
break
a1=a
b1=b
while b!=0:
c=a%b
a=b
b=c
print(a, a1*b1//a)
main()
``` | output | 1 | 77,476 | 22 | 154,953 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,477 | 22 | 154,954 |
"Correct Solution:
```
while True:
try:
a, b = map(int, input().split())
x, y = a, b
while y:
x, y = y, x%y
print(x,int(a*b/x))
except:
break
``` | output | 1 | 77,477 | 22 | 154,955 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,478 | 22 | 154,956 |
"Correct Solution:
```
#!/usr/bin/env python3
import sys
from fractions import gcd
for line in sys.stdin:
[a, b] = [int(x) for x in line.split()]
print(int(gcd(a, b)), int(a / gcd(a, b) * b))
``` | output | 1 | 77,478 | 22 | 154,957 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,479 | 22 | 154,958 |
"Correct Solution:
```
def gcd(a, b):
if b == 0: return a
else: return gcd(b, a % b)
def lcm(a, b):
return a * b / gcd(a, b)
while True:
try:
a, b = map(int, input().split())
print(int(gcd(a, b)), int(lcm(a, b)))
except EOFError:
break
``` | output | 1 | 77,479 | 22 | 154,959 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,480 | 22 | 154,960 |
"Correct Solution:
```
try:
while True:
x=list(sorted(map(int, input().split())))
a=x[0]
b=x[1]
while b%a != 0:
c=a
a=b%a
b=c
print(a, x[0]*x[1]//a)
except EOFError:
pass
``` | output | 1 | 77,480 | 22 | 154,961 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,481 | 22 | 154,962 |
"Correct Solution:
```
def gcd(a,b):
while b != 0:
a , b = b , a % b
return a
def lcm(a,b):
return int(abs(a*b) / gcd(a,b))
while True:
try:
a, b = [int(x) for x in input().split()]
except:
exit()
print(gcd(a,b),lcm(a,b))
``` | output | 1 | 77,481 | 22 | 154,963 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,482 | 22 | 154,964 |
"Correct Solution:
```
import sys
sys.setrecursionlimit(10**7)
import fileinput
def gcd(a, b):
while b:
a, b = b, a % b
return a
def lcm(a, b):
return a * b // gcd(a, b)
for line in sys.stdin:
x, y = map(int, line.split())
print(gcd(x,y),lcm(x,y))
``` | output | 1 | 77,482 | 22 | 154,965 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000 | instruction | 0 | 77,483 | 22 | 154,966 |
"Correct Solution:
```
import sys
def gcd(a, b):
while b % a:
a, b = b % a, a
return a
def lcm(a, b):
return a * b // gcd(a, b)
for line in sys.stdin:
a, b = sorted(list(map(int, line.split())))
print(gcd(a, b), lcm(a, b))
``` | output | 1 | 77,483 | 22 | 154,967 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
import sys
def gcd(a,b):
r= b % a
while r != 0:
a,b = r,a
r = b % a
return a
def lcm(a,b):
return int(a*b/gcd(a,b))
for line in sys.stdin:
a,b = sorted(map(int, line.rstrip().split(' ')))
print("{} {}".format(gcd(a,b),lcm(a,b)))
``` | instruction | 0 | 77,484 | 22 | 154,968 |
Yes | output | 1 | 77,484 | 22 | 154,969 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
def GCD_cal(a,b):
if(b==0):
return(a)
a,b=b,a%b
return(GCD_cal(a,b))
while(True):
try:
a,b=map(int,input().split(" "))
except:
break
if(a<b):
a,b=b,a
GCD=GCD_cal(a,b)
LCM=int(a*b/GCD)
print("{0} {1}".format(GCD,LCM))
``` | instruction | 0 | 77,485 | 22 | 154,970 |
Yes | output | 1 | 77,485 | 22 | 154,971 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
def gcd(li):
a=max(li)
b=min(li)
while b>0:
a,b=b,a%b
return a
def lcm(li):
return li[0]*li[1]/gcd(li)
while True:
try:
li=[int(i) for i in input().split(" ")]
print("%i %i"%(gcd(li),lcm(li)))
except:break
``` | instruction | 0 | 77,486 | 22 | 154,972 |
Yes | output | 1 | 77,486 | 22 | 154,973 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
def GCD(a, b):
if b == 0:
return a
return GCD(b, a % b)
def LCM(a, b):
return a * b // GCD(a, b)
import sys
s = sys.stdin.readlines()
n = len(s)
for i in range(n):
x, y = map(int, s[i].split())
print(GCD(x, y), LCM(x, y))
``` | instruction | 0 | 77,487 | 22 | 154,974 |
Yes | output | 1 | 77,487 | 22 | 154,975 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
import sys
def g(a, b):
d = abs(a - b)
return g(d, a) if a % d != 0 else d
def lcm(inta, intb, intgcd):
return (inta * intb // intgcd)
sets = sys.stdin.readlines()
for line in sets:
a, b = map(int, line.split())
c = g(a, b)
print(c, lcm(a,b,c))
``` | instruction | 0 | 77,488 | 22 | 154,976 |
No | output | 1 | 77,488 | 22 | 154,977 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
a,b=map(int,input().split())
for i in range(1,a+1):
f=(b*i)%a
lcm=(b*i)
if f==0:
break
for j in range(1,a+1):
if a%j==0 and b%j==0 and j*lcm==a*b:
print(j,lcm)
``` | instruction | 0 | 77,489 | 22 | 154,978 |
No | output | 1 | 77,489 | 22 | 154,979 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
import sys
for e in sys.stdin:
a, b = map(int, e.split())
p = a * b
while b:
a, b = b, a%b
pass
print (str(a)+' '+str(p/a))
``` | instruction | 0 | 77,490 | 22 | 154,980 |
No | output | 1 | 77,490 | 22 | 154,981 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which computes the greatest common divisor (GCD) and the least common multiple (LCM) of given a and b.
Constraints
* 0 < a, b ≤ 2,000,000,000
* LCM(a, b) ≤ 2,000,000,000
* The number of data sets ≤ 50
Input
Input consists of several data sets. Each data set contains a and b separated by a single space in a line. The input terminates with EOF.
Output
For each data set, print GCD and LCM separated by a single space in a line.
Example
Input
8 6
50000000 30000000
Output
2 24
10000000 150000000
Submitted Solution:
```
while 1 == 1:
a, b = map(int, input().split())
d = a * b
if a < b:
tmp = a
a = b
b = tmp
c = a % b
while(c != 0):
a = b
b = c
c = a % b
g = b
l = d / g
print(str(g)+" "+str(int(l)))
``` | instruction | 0 | 77,491 | 22 | 154,982 |
No | output | 1 | 77,491 | 22 | 154,983 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,657 | 22 | 155,314 |
Tags: brute force, math, number theory
Correct Solution:
```
import sys
input=sys.stdin.readline
import math
def gcd(a, b):
if a == 0 :
return b
return gcd(b%a, a)
t=int(input())
m=1000000000000
tup=[]
for i in range(1,int(t**0.5)+1):
if(t%i==0):
j=t//i
if(gcd(i,j)==1):
if(m>max(i,j)):
tup=[i,j]
m=max(i,j)
print(tup[0],tup[1])
``` | output | 1 | 77,657 | 22 | 155,315 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,658 | 22 | 155,316 |
Tags: brute force, math, number theory
Correct Solution:
```
a = int(input())
q = round(a**(1/2))
if q**2 > a:
q=q-1
def nod(a,b):
while a != 0 and b != 0:
if a > b:
a %= b
else:
b %= a
return a+b
while q > 0:
if a % q ==0:
p = a // q
if p*q//nod(p,q) == a:
print(q, p)
break
q = q-1
``` | output | 1 | 77,658 | 22 | 155,317 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,659 | 22 | 155,318 |
Tags: brute force, math, number theory
Correct Solution:
```
# |
# _` | __ \ _` | __| _ \ __ \ _` | _` |
# ( | | | ( | ( ( | | | ( | ( |
# \__,_| _| _| \__,_| \___| \___/ _| _| \__,_| \__,_|
import sys
import math
def read_line():
return sys.stdin.readline()[:-1]
def read_int():
return int(sys.stdin.readline())
def read_int_line():
return [int(v) for v in sys.stdin.readline().split()]
def read_float_line():
return [float(v) for v in sys.stdin.readline().split()]
def gcd(a,b):
if a == 0:
return b
return gcd(b % a, a)
def printDivisors(n):
i = 1
a,b = 0,0
while i <= math.sqrt(n):
if (n % i == 0) :
if gcd(i,n//i)==1:
a = i
b = n//i
i = i + 1
return (a,b)
x =read_int()
ans = printDivisors(x)
print(*ans)
``` | output | 1 | 77,659 | 22 | 155,319 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,660 | 22 | 155,320 |
Tags: brute force, math, number theory
Correct Solution:
```
import math
n=int(input())
ans=n
for i in range(1,int(math.sqrt(n))+1):
if n%i==0:
a=i
b=n//i
if math.gcd(i,n//i)==1:
ans=min(ans,n//i)
print(n//ans,ans)
``` | output | 1 | 77,660 | 22 | 155,321 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,661 | 22 | 155,322 |
Tags: brute force, math, number theory
Correct Solution:
```
from fractions import gcd
def main():
X = int(input())
now = [10**20,10**20]
for i in range(1,int(X**(0.5))+1):
if X%i==0 and X==(i*(X//i))//gcd(i,X//i):
if max(i,X//i)<max(now):
now = [i,X//i]
print(now[0],now[1])
if __name__ == '__main__':
main()
``` | output | 1 | 77,661 | 22 | 155,323 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,662 | 22 | 155,324 |
Tags: brute force, math, number theory
Correct Solution:
```
#from statistics import median
#import collections
#aa = collections.Counter(a) # list to list || .most_common(2)で最大の2個とりだせるお a[0][0]
# from math import gcd
# from itertools import combinations,permutations,accumulate, product # (string,3) 3回
# #from collections import deque
# from collections import deque,defaultdict,Counter
# import decimal
# import re
# import math
# import bisect
# import heapq
#
#
#
# pythonで無理なときは、pypyでやると正解するかも！！
#
#
# my_round_int = lambda x:np.round((x*2 + 1)//2)
# 四捨五入g
#
# インデックス系
# int min_y = max(0, i - 2), max_y = min(h - 1, i + 2);
# int min_x = max(0, j - 2), max_x = min(w - 1, j + 2);
#
#
import sys
sys.setrecursionlimit(10000000)
mod = 10**9 + 7
#mod = 9982443453
#mod = 998244353
INF = float('inf')
from sys import stdin
readline = stdin.readline
def readInts():
return list(map(int,readline().split()))
def readTuples():
return tuple(map(int,readline().split()))
def I():
return int(readline())
from math import gcd
def yaku(m):
ans = []
i = 1
while i*i <= m:
if m % i == 0:
j = m // i
ans.append(i)
if j != i:
ans.append(j)
i += 1
ans = sorted(ans)
return ans
n = I()
lis = yaku(n)
#print(lis)
ln = len(lis)
#print((4 * 6)/gcd(4,6))
if n == 1:
print(1, 1)
elif ln == 2:
print(1, lis[-1])
else:
if ln%2:
cnt = 0
while cnt < ln//2:
if ((lis[ln//2 + cnt + 1] * lis[ln//2 - cnt -1]) // gcd(lis[ln//2 + cnt + 1], lis[ln//2 - cnt - 1])) == n:
print(lis[ln//2 + cnt + 1], lis[ln//2 - cnt - 1])
break
else:
cnt += 1
else:
cnt = 0
while cnt < ln//2:
if ((lis[ln//2 + cnt] * lis[ln//2 - cnt - 1]) // gcd(lis[ln//2 + cnt], lis[ln//2 - cnt - 1])) == n:
print(lis[ln//2 + cnt], lis[ln//2 - cnt - 1])
break
else:
cnt += 1
``` | output | 1 | 77,662 | 22 | 155,325 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,663 | 22 | 155,326 |
Tags: brute force, math, number theory
Correct Solution:
```
from sys import maxsize, stdout, stdin,stderr
mod = int(1e9 + 7)
def I(): return int(stdin.readline())
def lint(): return [int(x) for x in stdin.readline().split()]
def S(): return input().strip()
def grid(r, c): return [lint() for i in range(r)]
from collections import defaultdict, Counter
import math
from itertools import groupby
def gcd(a,b):
while b:
a %= b
tmp = a
a = b
b = tmp
return a
def lcm(a,b):
return a / gcd(a, b) * b
def check_prime(n):
for i in range(2,n):
if n%i==0:
return 0
return 1
n = I()
ans=None
i=1
while i**2<=n:
if n%i==0 and lcm(i, n//i)==n:
ans=i
i+=1
print(ans, n//ans)
``` | output | 1 | 77,663 | 22 | 155,327 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1 | instruction | 0 | 77,664 | 22 | 155,328 |
Tags: brute force, math, number theory
Correct Solution:
```
#python3
def gcd(a,b):
if b==0:
return a
else:
return gcd(b,a%b)
def lcm(a,b):
return (a*b)/gcd(a,b)
#print(gcd(16,24),lcm(6,8))
def fadi(X):
l=[] #list of pairs whose lcm is X
i=2
a=1
b=X
while i*i<=X:
if X%i==0 and lcm(i,X/i)==X:
a=i
b=X/i
i+=1
else:
i+=1
print(str(int(a))+" "+str(int(b)))
fadi(int(input()))
``` | output | 1 | 77,664 | 22 | 155,329 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
from math import sqrt,gcd
n=int(input())
l=int(sqrt(n))
for i in range(l,-1,-1):
if n%i==0 and gcd(i,n//i)==1: break
print(i,n//i)
``` | instruction | 0 | 77,665 | 22 | 155,330 |
Yes | output | 1 | 77,665 | 22 | 155,331 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
import sys
input = sys.stdin.readline
x=int(input())
y=x
ANS=x+1
AX=[0,0]
import math
L=int(math.sqrt(x))
FACT=dict()
for i in range(2,L+2):
while x%i==0:
FACT[i]=FACT.get(i,0)+1
x=x//i
if x!=1:
FACT[x]=FACT.get(x,0)+1
x=y
LEN=len(FACT)
LIST=list(FACT.keys())
for i in range(1<<LEN):
sc=1
for j in range(LEN):
if i & (1<<j) !=0:
sc*=LIST[j]**FACT[LIST[j]]
if ANS>max(sc,x//sc):
ANS=max(sc,x//sc)
AX=[sc,x//sc]
print(*AX)
``` | instruction | 0 | 77,666 | 22 | 155,332 |
Yes | output | 1 | 77,666 | 22 | 155,333 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
import math
x=int(input())
a=b=x+1
n=int(math.sqrt(x))+1
for i in range(1,n):
if ((x%i)==0):
y=x//i
if (i*y==math.gcd(i,y)*x):
if (y<b):
a=i
b=y
print(a,b)
``` | instruction | 0 | 77,667 | 22 | 155,334 |
Yes | output | 1 | 77,667 | 22 | 155,335 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
''' Hey stalker :) '''
INF = 10**10
def main():
#print = out.append
''' Cook your dish here! '''
n = get_int()
for x in range(int(n**0.5), 0, -1):
if n%x==0 and math.gcd(x, n//x)==1:
print(x, n//x)
return
''' Pythonista fLite 1.1 '''
import sys
#from collections import defaultdict, Counter
#from functools import reduce
import math
#input = iter(sys.stdin.buffer.read().decode().splitlines()).__next__
out = []
get_int = lambda: int(input())
get_list = lambda: list(map(int, input().split()))
main()
#[main() for _ in range(int(input()))]
#print(*out, sep='\n')
``` | instruction | 0 | 77,668 | 22 | 155,336 |
Yes | output | 1 | 77,668 | 22 | 155,337 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
import math
n = int(input())
def primeFactors(n):
ans = []
# Print the number of two's that divide n
fac = 1
while n % 2 == 0:
n = n // 2
fac *= 2
ans.append(fac)
for i in range(3,int(math.sqrt(n))+1,2):
fac = 1
while n % i== 0:
n = n // i
fac *= i
ans.append(fac)
if n > 2:
ans.append(n)
return ans
def dfs(prod, ind):
global ma
if ind == len(ans):
return
if prod > li:
return
ma = max(ma, prod)
dfs(prod * ans[ind], ind + 1)
dfs(prod, ind + 1)
ans = primeFactors(n)
li = math.floor(math.sqrt(n))
ma = 1
dfs(1, 0)
print(ma, n//ma)
``` | instruction | 0 | 77,669 | 22 | 155,338 |
No | output | 1 | 77,669 | 22 | 155,339 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
Input
The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
Output
Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
Examples
Input
2
Output
1 2
Input
6
Output
2 3
Input
4
Output
1 4
Input
1
Output
1 1
Submitted Solution:
```
from itertools import chain, combinations
from operator import mul
from functools import reduce
import sys
def primes(n):
primfac = [1]
d = 2
while d*d <= n:
while (n % d) == 0:
if primfac[-1] % d!=0:
primfac.append(d) # supposing you want multiple factors repeated
else:
primfac[-1]*=d
n //= d
d += 1
if n > 1:
primfac.append(n)
return primfac
X = int(input())
ps = list(sorted(primes(X)[1:]))
ans = sys.maxsize
amin=1
bmin=1
def rec(i,j,l):
if l == 0:
global ans,amin,bmin
if max(i,j)<ans:
ans=max(i,j)
amin=i
bmin=j
return
rec(i*ps[-l],j,l-1)
rec(i,j*ps[-l],l-1)
rec(1,1,len(ps))
print(amin,bmin)
``` | instruction | 0 | 77,670 | 22 | 155,340 |
No | output | 1 | 77,670 | 22 | 155,341 |
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