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 two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,423 | 22 | 68,846 |
Tags: math
Correct Solution:
```
t = int(input())
for i in range(t):
a,b = map(int,input().split())
if a%b == 0:
print(0)
else:
m = a // b
print((b*(m+1)) - a)
``` | output | 1 | 34,423 | 22 | 68,847 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,424 | 22 | 68,848 |
Tags: math
Correct Solution:
```
t=int(input())
for i in range(0,t):
y=input().split()
a=int(y[0])
b=int(y[1])
m=a%b
if m==0:
print(m)
else:
print(b-m)
``` | output | 1 | 34,424 | 22 | 68,849 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,425 | 22 | 68,850 |
Tags: math
Correct Solution:
```
def incrementation(a,b):
if a % b == 0:
return 0
else:
r = a % b
return b-r
testCases = int(input())
for i in range(testCases):
a,b = map(int,input().split())
print(incrementation(a,b))
``` | output | 1 | 34,425 | 22 | 68,851 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,426 | 22 | 68,852 |
Tags: math
Correct Solution:
```
t=int(input())
while t>0:
t-=1
a,b=map(int,input().split())
c=a//b
d=a%b
if d==0:
print(0)
else:
print((c+1)*b-a)
``` | output | 1 | 34,426 | 22 | 68,853 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,427 | 22 | 68,854 |
Tags: math
Correct Solution:
```
for i in range(int(input())):
a,b = map(int,input().split())
if a%b == 0 :
print(0)
else:
print(b - a % b)
``` | output | 1 | 34,427 | 22 | 68,855 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,428 | 22 | 68,856 |
Tags: math
Correct Solution:
```
n = int(input())
steps = []
for i in range(n):
a, b = map(int, input().split())
if a%b != 0:
step = (((a//b)+1)*b)-a
steps.append(step)
else:
steps.append(0)
for i in steps: print(i)
``` | output | 1 | 34,428 | 22 | 68,857 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0 | instruction | 0 | 34,429 | 22 | 68,858 |
Tags: math
Correct Solution:
```
for t in range(int(input())):
a,b=map(int,input().split())
if a<=b:
print(b-a)
else:
if a%b==0:
print(0)
else:
v=a//b+1
print(b*v-a)
``` | output | 1 | 34,429 | 22 | 68,859 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
t = int(input())
matrix = []
answer_list = []
for _ in range(t):
matrix.append(list(map(int, input().split())))
for row in range(len(matrix)):
for col in range(1, len(matrix[row])):
if matrix[row][col-1] % matrix[row][col] != 0:
remainder = matrix[row][col-1] % matrix[row][col]
answer = matrix[row][col]-remainder
answer_list.append(answer)
elif matrix[row][col-1] % matrix[row][col] == 0:
answer = matrix[row][col-1] % matrix[row][col]
answer_list.append(answer)
for i in answer_list:
print(i)
``` | instruction | 0 | 34,430 | 22 | 68,860 |
Yes | output | 1 | 34,430 | 22 | 68,861 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
for _ in range(int(input())):
a,b=map(int,input().split())
if(a%b==0):
print(0)
else:
n=a//b
m=(a//b)*b+b-a
print(m)
``` | instruction | 0 | 34,431 | 22 | 68,862 |
Yes | output | 1 | 34,431 | 22 | 68,863 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
results = []
for _ in range(int(input())):
a, b = map(int, input().split())
if a % b:
results.append(str(b - a % b))
else:
results.append('0')
print('\n'.join(results))
``` | instruction | 0 | 34,432 | 22 | 68,864 |
Yes | output | 1 | 34,432 | 22 | 68,865 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
t=int(input())
for i in range(t):
a,b=map(int,input().split())
if(a%b==0):
print('0')
else:
print(b-(a%b))
``` | instruction | 0 | 34,433 | 22 | 68,866 |
Yes | output | 1 | 34,433 | 22 | 68,867 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
t=int(input())
while t>0:
t-=1
n,k=input().split()
n,k=int(n),int(k)
if n>=k and n%2==k%2:
print("YES")
else:
print("NO")
``` | instruction | 0 | 34,434 | 22 | 68,868 |
No | output | 1 | 34,434 | 22 | 68,869 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
import sys
def get_array(): return list(map(int , sys.stdin.readline().strip().split()))
def get_ints(): return map(int, sys.stdin.readline().strip().split())
def input(): return sys.stdin.readline().strip()
for _ in range(int(input())):
a,b=get_ints()
print(a%b)
``` | instruction | 0 | 34,435 | 22 | 68,870 |
No | output | 1 | 34,435 | 22 | 68,871 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
for i in range(int(input())):
n,k = map(int,input().split())
if n%k == 0 :
print(0)
else:
print(n-(n%k))
``` | instruction | 0 | 34,436 | 22 | 68,872 |
No | output | 1 | 34,436 | 22 | 68,873 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two positive integers a and b. In one move you can increase a by 1 (replace a with a+1). Your task is to find the minimum number of moves you need to do in order to make a divisible by b. It is possible, that you have to make 0 moves, as a is already divisible by b. You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers a and b (1 β€ a, b β€ 10^9).
Output
For each test case print the answer β the minimum number of moves you need to do in order to make a divisible by b.
Example
Input
5
10 4
13 9
100 13
123 456
92 46
Output
2
5
4
333
0
Submitted Solution:
```
def main(a,b):
a0 = a
for i in range(a,a+11):
if not a%b:
break
else:
a+=1
return a-a0
if __name__=='__main__':
n = int(input())
for i in range(n):
a,b = list(map(int,input().split()))
print(main(a, b))
``` | instruction | 0 | 34,437 | 22 | 68,874 |
No | output | 1 | 34,437 | 22 | 68,875 |
Provide tags and a correct Python 3 solution for this coding contest problem.
JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer n, you can perform the following operations zero or more times:
* mul x: multiplies n by x (where x is an arbitrary positive integer).
* sqrt: replaces n with β{n} (to apply this operation, β{n} must be an integer).
You can perform these operations as many times as you like. What is the minimum value of n, that can be achieved and what is the minimum number of operations, to achieve that minimum value?
Apparently, no one in the class knows the answer to this problem, maybe you can help them?
Input
The only line of the input contains a single integer n (1 β€ n β€ 10^6) β the initial number.
Output
Print two integers: the minimum integer n that can be achieved using the described operations and the minimum number of operations required.
Examples
Input
20
Output
10 2
Input
5184
Output
6 4
Note
In the first example, you can apply the operation mul 5 to get 100 and then sqrt to get 10.
In the second example, you can first apply sqrt to get 72, then mul 18 to get 1296 and finally two more sqrt and you get 6.
Note, that even if the initial value of n is less or equal 10^6, it can still become greater than 10^6 after applying one or more operations. | instruction | 0 | 35,156 | 22 | 70,312 |
Tags: greedy, math, number theory
Correct Solution:
```
import math
def list_primes_upto(upto :int):
uptop1 = upto+1
uptom1 = upto-1
pcl = [True for x in range(uptop1)]
lprimes = []
if upto >= 2: lprimes.append(2)
if upto >= 3: lprimes.append(3)
i = 5
while i < uptom1:
ix2 = i*2
if pcl[i]:
lprimes.append(i)
for k in range(i*i, uptop1, ix2):
pcl[k] = False
ip2 = i+2
ip2x2 = ip2*2
if pcl[ip2]:
lprimes.append(ip2)
for k in range(ip2*ip2, uptop1, ip2x2):
pcl[k] = False
i += 6
if i <= upto:
ix2 = i*2
if pcl[i]:
lprimes.append(i)
for k in range(i*i, uptop1, ix2):
pcl[k] = False
return lprimes
n0 = int(input())
n = n0
lp = list_primes_upto(int(math.sqrt(n)) + 1)
ncp = 0
pmf = [0] * len(lp)
while ncp < len(lp) and n != 1:
if n % lp[ncp] != 0:
ncp += 1
continue
n //= lp[ncp]
pmf[ncp] += 1
mn = n
sf = set(pmf)
if n != 1:
sf.add(1)
for i in range(len(lp)):
if pmf[i] == 0:
continue
mn *= lp[i]
if mn == n0:
mo = 0
else:
mpf = max(sf)
impfl2 = int(math.log(mpf, 2))
ls2 = [2**x for x in range(0,1001)]
ntm = 0
for f in sf:
if f == 0:
continue
if f not in ls2 or f != mpf:
ntm = 1
break
if mpf in ls2:
sro = impfl2
else:
sro = impfl2 + 1
ntm = 1
mo = ntm + sro
print(mn, mo)
``` | output | 1 | 35,156 | 22 | 70,313 |
Provide tags and a correct Python 3 solution for this coding contest problem.
JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer n, you can perform the following operations zero or more times:
* mul x: multiplies n by x (where x is an arbitrary positive integer).
* sqrt: replaces n with β{n} (to apply this operation, β{n} must be an integer).
You can perform these operations as many times as you like. What is the minimum value of n, that can be achieved and what is the minimum number of operations, to achieve that minimum value?
Apparently, no one in the class knows the answer to this problem, maybe you can help them?
Input
The only line of the input contains a single integer n (1 β€ n β€ 10^6) β the initial number.
Output
Print two integers: the minimum integer n that can be achieved using the described operations and the minimum number of operations required.
Examples
Input
20
Output
10 2
Input
5184
Output
6 4
Note
In the first example, you can apply the operation mul 5 to get 100 and then sqrt to get 10.
In the second example, you can first apply sqrt to get 72, then mul 18 to get 1296 and finally two more sqrt and you get 6.
Note, that even if the initial value of n is less or equal 10^6, it can still become greater than 10^6 after applying one or more operations. | instruction | 0 | 35,158 | 22 | 70,316 |
Tags: greedy, math, number theory
Correct Solution:
```
import math
def primeFactors(n):
if n%2==0:
dict1[2]=0
while n % 2 == 0:
dict1[2]+=1
n = n // 2
for i in range(3,int(math.sqrt(n))+1,2):
if n%i==0:
dict1[i]=0
while n % i== 0:
dict1[i]+=1
n = n //i
if n > 2:
dict1[n]=0
dict1[n]+=1
n=int(input())
if n==1:
print(1,0)
else:
dict1={}
primeFactors(n)
max1=1
lst=[]
for i in dict1:
lst.append(dict1[i])
if max(lst)==min(lst) and max(lst)==1:
print(n,0)
elif max(lst)==min(lst):
prod=1
for i in dict1:
prod*=i
str1=bin(max(lst))
str2=str1[2:]
list1=list(str2)
if list1.count('0')==len(list1)-1:
print(prod,math.ceil(math.log2(max(lst))))
else:
print(prod,math.ceil(math.log2(max(lst)))+1)
else:
prod=1
for i in dict1:
prod*=i
print(prod,math.ceil(math.log2(max(lst)))+1)
``` | output | 1 | 35,158 | 22 | 70,317 |
Provide tags and a correct Python 3 solution for this coding contest problem.
JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer n, you can perform the following operations zero or more times:
* mul x: multiplies n by x (where x is an arbitrary positive integer).
* sqrt: replaces n with β{n} (to apply this operation, β{n} must be an integer).
You can perform these operations as many times as you like. What is the minimum value of n, that can be achieved and what is the minimum number of operations, to achieve that minimum value?
Apparently, no one in the class knows the answer to this problem, maybe you can help them?
Input
The only line of the input contains a single integer n (1 β€ n β€ 10^6) β the initial number.
Output
Print two integers: the minimum integer n that can be achieved using the described operations and the minimum number of operations required.
Examples
Input
20
Output
10 2
Input
5184
Output
6 4
Note
In the first example, you can apply the operation mul 5 to get 100 and then sqrt to get 10.
In the second example, you can first apply sqrt to get 72, then mul 18 to get 1296 and finally two more sqrt and you get 6.
Note, that even if the initial value of n is less or equal 10^6, it can still become greater than 10^6 after applying one or more operations. | instruction | 0 | 35,159 | 22 | 70,318 |
Tags: greedy, math, number theory
Correct Solution:
```
# Python program to print prime factors
import math
def primeFactors(n):
# Print the number of two's that divide n
dic = {}
while n % 2 == 0:
if 2 in dic:
dic[2] += 1
else:
dic[2] = 1
n = n / 2
# n must be odd at this point
# so a skip of 2 ( i = i + 2) can be used
for i in range(3,int(math.sqrt(n))+1,2):
# while i divides n , print i ad divide n
while n % i== 0:
if i in dic:
dic[i] += 1
else:
dic[i] = 1
n = n / i
# Condition if n is a prime
# number greater than 2
if n > 2:
if n in dic:
dic[n] += 1
else:
dic[n] = 1
return dic
n = int(input())
arr = primeFactors(n)
arr2 = []
for key in arr:
arr2.append(arr[key])
length = len(arr2)
ans = 0
#print("ura")
while True:
sqrt = True
if len(arr2) == 0:
break
for i in arr2:
if i%2 == 1:
sqrt = False
break
if sqrt:
ans += 1
for i in range(length):
arr2[i] = arr2[i]/2
else:
break
p = 0
start = 1
if len(arr2) ==0:
a = 1
else:
a = max(arr2)
for i in range(1000):
if start<a:
start *= 2
p += 1
else:
break
plus1 = False
for i in range(length):
if arr2[i] <start:
plus1 = True
break
if plus1:
ans += p + 1
else:
ans += p
ans0 = 1
for i in arr:
ans0 *= i
print(int(ans0),ans)
``` | output | 1 | 35,159 | 22 | 70,319 |
Provide tags and a correct Python 3 solution for this coding contest problem.
JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer n, you can perform the following operations zero or more times:
* mul x: multiplies n by x (where x is an arbitrary positive integer).
* sqrt: replaces n with β{n} (to apply this operation, β{n} must be an integer).
You can perform these operations as many times as you like. What is the minimum value of n, that can be achieved and what is the minimum number of operations, to achieve that minimum value?
Apparently, no one in the class knows the answer to this problem, maybe you can help them?
Input
The only line of the input contains a single integer n (1 β€ n β€ 10^6) β the initial number.
Output
Print two integers: the minimum integer n that can be achieved using the described operations and the minimum number of operations required.
Examples
Input
20
Output
10 2
Input
5184
Output
6 4
Note
In the first example, you can apply the operation mul 5 to get 100 and then sqrt to get 10.
In the second example, you can first apply sqrt to get 72, then mul 18 to get 1296 and finally two more sqrt and you get 6.
Note, that even if the initial value of n is less or equal 10^6, it can still become greater than 10^6 after applying one or more operations. | instruction | 0 | 35,161 | 22 | 70,322 |
Tags: greedy, math, number theory
Correct Solution:
```
def main():
buf = input()
n = int(buf)
op_count = 0
factor = prime_factorization(n)
c_max = 1
first_c = None
all_same = True
min_number = 1
for f, c in factor.items():
if c > c_max:
c_max = c
if first_c == None:
first_c = c
elif c != first_c:
all_same = False
min_number *= f
if check_power_of_2(c_max) and all_same:
pass
else:
op_count += 1
if not check_power_of_2(c_max):
c_max = c_max << 1
c_max = c_max & (c_max - 1)
while c_max > 1:
c_max //= 2
op_count += 1
print(min_number, op_count)
def check_power_of_2(x):
return (x != 0) and ((x & (x - 1)) == 0)
def prime_factorization(number):
n = number
i = 2
factor = {}
while i * i <= n:
while n % i == 0:
if i in factor:
factor[i] += 1
else:
factor.update({i : 1})
n //= i
i += 1
if n > 1 or not factor:
if n in factor:
factor[n] += 1
else:
factor.update({n : 1})
return factor
if __name__ == '__main__':
main()
``` | output | 1 | 35,161 | 22 | 70,323 |
Provide tags and a correct Python 3 solution for this coding contest problem.
JATC's math teacher always gives the class some interesting math problems so that they don't get bored. Today the problem is as follows. Given an integer n, you can perform the following operations zero or more times:
* mul x: multiplies n by x (where x is an arbitrary positive integer).
* sqrt: replaces n with β{n} (to apply this operation, β{n} must be an integer).
You can perform these operations as many times as you like. What is the minimum value of n, that can be achieved and what is the minimum number of operations, to achieve that minimum value?
Apparently, no one in the class knows the answer to this problem, maybe you can help them?
Input
The only line of the input contains a single integer n (1 β€ n β€ 10^6) β the initial number.
Output
Print two integers: the minimum integer n that can be achieved using the described operations and the minimum number of operations required.
Examples
Input
20
Output
10 2
Input
5184
Output
6 4
Note
In the first example, you can apply the operation mul 5 to get 100 and then sqrt to get 10.
In the second example, you can first apply sqrt to get 72, then mul 18 to get 1296 and finally two more sqrt and you get 6.
Note, that even if the initial value of n is less or equal 10^6, it can still become greater than 10^6 after applying one or more operations. | instruction | 0 | 35,162 | 22 | 70,324 |
Tags: greedy, math, number theory
Correct Solution:
```
def factorize(n):
fct = [] # prime factor
b, e = 2, 0 # base, exponent
while b * b <= n:
while n % b == 0:
n = n // b
e = e + 1
if e > 0:
fct.append((b, e))
b, e = b + 1, 0
if n > 1:
fct.append((n, 1))
return fct
def main():
n = int(input())
l = factorize(n)
anscnt = 0
ma = 0
for i in range(len(l)):
ma = max(ma, l[i][1])
c = 0
a = 1
while True:
if a >= ma:
break
a *= 2
c += 1
v = 1
for i in range(len(l)):
for j in range(a - l[i][1]):
v *= l[i][0]
anscnt = 1
n *= v
for i in range(c):
n = n**(1/2)
anscnt += 1
print(int(n), anscnt, sep=" ")
if __name__ == '__main__':
main()
``` | output | 1 | 35,162 | 22 | 70,325 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
If an integer a is divisible by another integer b, then b is called the divisor of a.
For example: 12 has positive 6 divisors. They are 1, 2, 3, 4, 6 and 12.
Letβs define a function D(n) β number of integers between 1 and n (inclusive) which has exactly four positive divisors.
Between 1 and 10 only the integers 6, 8 and 10 has exactly four positive divisors. So, D(10) = 3.
You are given an integer n. You have to calculate D(n).
Input
The only line contains integer n (1 β€ n β€ 1011) β the parameter from the problem statement.
Output
Print the only integer c β the number of integers between 1 and n with exactly four divisors.
Examples
Input
10
Output
3
Input
20
Output
5
Submitted Solution:
```
n=int(input(""))
if n==1:
print("I hate it")
else:
for i in range(1,n):
if i%2==0:
print("I love that",end=" ")
else:
print("I hate that",end=" ")
if n%2==0:
print("")
else:
print("")
#https://codeforces.com/problemset/problem/705/A
``` | instruction | 0 | 35,608 | 22 | 71,216 |
No | output | 1 | 35,608 | 22 | 71,217 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
If an integer a is divisible by another integer b, then b is called the divisor of a.
For example: 12 has positive 6 divisors. They are 1, 2, 3, 4, 6 and 12.
Letβs define a function D(n) β number of integers between 1 and n (inclusive) which has exactly four positive divisors.
Between 1 and 10 only the integers 6, 8 and 10 has exactly four positive divisors. So, D(10) = 3.
You are given an integer n. You have to calculate D(n).
Input
The only line contains integer n (1 β€ n β€ 1011) β the parameter from the problem statement.
Output
Print the only integer c β the number of integers between 1 and n with exactly four divisors.
Examples
Input
10
Output
3
Input
20
Output
5
Submitted Solution:
```
def D(n):
for i in range(1,n+1):
c = 0
for j in range(1, i):
if(i%j==0):
c=c+1
print(c)
n = int(input())
D(n)
``` | instruction | 0 | 35,609 | 22 | 71,218 |
No | output | 1 | 35,609 | 22 | 71,219 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
If an integer a is divisible by another integer b, then b is called the divisor of a.
For example: 12 has positive 6 divisors. They are 1, 2, 3, 4, 6 and 12.
Letβs define a function D(n) β number of integers between 1 and n (inclusive) which has exactly four positive divisors.
Between 1 and 10 only the integers 6, 8 and 10 has exactly four positive divisors. So, D(10) = 3.
You are given an integer n. You have to calculate D(n).
Input
The only line contains integer n (1 β€ n β€ 1011) β the parameter from the problem statement.
Output
Print the only integer c β the number of integers between 1 and n with exactly four divisors.
Examples
Input
10
Output
3
Input
20
Output
5
Submitted Solution:
```
import sys
def divi(n,k):
if k==1:
return 0
elif n%k==0:
return 1+divi(n,k-1)
else:
return 0+divi(n,k-1)
n=int(sys.stdin.readline().strip())
print(divi(n,n))
``` | instruction | 0 | 35,610 | 22 | 71,220 |
No | output | 1 | 35,610 | 22 | 71,221 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
If an integer a is divisible by another integer b, then b is called the divisor of a.
For example: 12 has positive 6 divisors. They are 1, 2, 3, 4, 6 and 12.
Letβs define a function D(n) β number of integers between 1 and n (inclusive) which has exactly four positive divisors.
Between 1 and 10 only the integers 6, 8 and 10 has exactly four positive divisors. So, D(10) = 3.
You are given an integer n. You have to calculate D(n).
Input
The only line contains integer n (1 β€ n β€ 1011) β the parameter from the problem statement.
Output
Print the only integer c β the number of integers between 1 and n with exactly four divisors.
Examples
Input
10
Output
3
Input
20
Output
5
Submitted Solution:
```
n = input()
n = int(n)
tw = 0
th = 0
fi = 0
while(n > 1):
if((n % 2) == 0):
n = n/2
tw = tw + 1
elif((n % 3) == 0):
n = n/3
th = th + 1
elif((n % 5) == 0):
n = n/5
fi = fi + 1
div = (tw + 1)*(th + 1)*(fi + 1)
print(div)
``` | instruction | 0 | 35,611 | 22 | 71,222 |
No | output | 1 | 35,611 | 22 | 71,223 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,646 | 22 | 71,292 |
Tags: greedy, math, number theory
Correct Solution:
```
def prime_t(t):
i=2
while i**2<=t:
if t%i==0:
return 0
i+=1
return 1
def prime_list(tt):
p_list=[]
for i in range(2,tt+1):
if prime_t(i):
p_list.append(i)
return p_list
chk=prime_list(318)
ans=[0]*(len(chk)+1)
ans[-1]=1
n=int(input())
l=[int(i) for i in input().split() if i!='1']
d={}
for i in l:
if i in d:
d[i]+=1
ans[-1]=max(ans[-1],d[i])
else:
d[i]=1
l=list(set(l))
l.sort()
over={}
for i in l:
tmp=i
for j,k in enumerate(chk):
if i%k==0:
ans[j]+=d[tmp]
while i%k==0:
i//=k
if i<k:
break
if i>317:
if i in over:
over[i]+=d[tmp]
ans[-1]=max(ans[-1],over[i])
else:
over[i]=d[tmp]
print(max(ans))
``` | output | 1 | 35,646 | 22 | 71,293 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,647 | 22 | 71,294 |
Tags: greedy, math, number theory
Correct Solution:
```
from collections import Counter
newn = int(input())
unfs = list(map(int, input().split()))
s = []
n = 0
p = [2, 3]
for i in range(1, 55):
p.append(6*i - 1)
p.append(6*i + 1)
if newn == 1 and unfs[0] == 1:
print(1)
else:
for i in unfs:
if i != 1:
s.append(i)
n += 1
a = []
for i in range(n):
found = False
for j in range(1, 1+int(s[i]**0.5)):
if s[i]%j == 0:
if j != 1:
a.append(j)
if s[i]//j != 1 and s[i]//j != j:
a.append(s[i]//j)
c = Counter(a).most_common()
if len(c) == 0:
print(1)
else:
print(c[0][1])
``` | output | 1 | 35,647 | 22 | 71,295 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,648 | 22 | 71,296 |
Tags: greedy, math, number theory
Correct Solution:
```
n=int(input())
a=list(map(int,input().split()))
b=[0]*100001
c=[]
for i in a:
b[i]+=1
for i in range(2,100001):
sum=0
for j in range(i,100001,i):
sum+=b[j]
c.append(sum)
print(max(max(c),1))
``` | output | 1 | 35,648 | 22 | 71,297 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,649 | 22 | 71,298 |
Tags: greedy, math, number theory
Correct Solution:
```
n = int(input())
a = list(map(int, input().split()))
ans = 1
count = [0] * 100001
for i in a:
count[i] += 1
for i in range(2, 100001):
s = 0
for j in range(i, 100001, i):
s += count[j]
ans = max(ans, s)
print(ans)
``` | output | 1 | 35,649 | 22 | 71,299 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,650 | 22 | 71,300 |
Tags: greedy, math, number theory
Correct Solution:
```
n=int(input())
d={}
l=list(map(int,input().split()))
for i in range(n):
x=int(l[i]**0.5)+1
for j in range(2,x):
if l[i]%j==0:
d[j]=d.get(j,0)+1
while l[i]%j==0:
l[i]//=j
if l[i]>1:
d[l[i]]=d.get(l[i],0)+1
# print(d)
if d=={}:
print(1)
exit(0)
print(max(1,max(d.values())))
``` | output | 1 | 35,650 | 22 | 71,301 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Bash has set out on a journey to become the greatest Pokemon master. To get his first Pokemon, he went to Professor Zulu's Lab. Since Bash is Professor Zulu's favourite student, Zulu allows him to take as many Pokemon from his lab as he pleases.
But Zulu warns him that a group of k > 1 Pokemon with strengths {s1, s2, s3, ..., sk} tend to fight among each other if gcd(s1, s2, s3, ..., sk) = 1 (see notes for gcd definition).
Bash, being smart, does not want his Pokemon to fight among each other. However, he also wants to maximize the number of Pokemon he takes from the lab. Can you help Bash find out the maximum number of Pokemon he can take?
Note: A Pokemon cannot fight with itself.
Input
The input consists of two lines.
The first line contains an integer n (1 β€ n β€ 105), the number of Pokemon in the lab.
The next line contains n space separated integers, where the i-th of them denotes si (1 β€ si β€ 105), the strength of the i-th Pokemon.
Output
Print single integer β the maximum number of Pokemons Bash can take.
Examples
Input
3
2 3 4
Output
2
Input
5
2 3 4 6 7
Output
3
Note
gcd (greatest common divisor) of positive integers set {a1, a2, ..., an} is the maximum positive integer that divides all the integers {a1, a2, ..., an}.
In the first sample, we can take Pokemons with strengths {2, 4} since gcd(2, 4) = 2.
In the second sample, we can take Pokemons with strengths {2, 4, 6}, and there is no larger group with gcd β 1. | instruction | 0 | 35,651 | 22 | 71,302 |
Tags: greedy, math, number theory
Correct Solution:
```
n = int(input())
s = map(int, input().split())
pf = [[] for _ in range(10 ** 5 + 1)]
for i in range(2, 10 ** 5 + 1):
if len(pf[i]) > 0:
continue
for j in range(i, 10 ** 5 + 1, i):
pf[j].append(i)
f = [0] * (10 ** 5 + 1)
for x in s:
for i in pf[x]:
f[i] += 1
print(max(1, max(f)))
``` | output | 1 | 35,651 | 22 | 71,303 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,047 | 22 | 72,094 |
Tags: number theory
Correct Solution:
```
n = int(input())
c = [0] * 9
ans = 0
for i in range(1, n + 1):
ans -= n // i
c[i % 9] += 1
for i in range(9):
for j in range(9):
ans += c[i] * c[j] * c[i * j % 9]
print(ans)
``` | output | 1 | 36,047 | 22 | 72,095 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,048 | 22 | 72,096 |
Tags: number theory
Correct Solution:
```
n, ans, a = int(input()), 0, [0] * 10
for i in range(1, n + 1):
ans -= int(n/i)
a[i % 9] += 1
for i in range(9):
for j in range(9):
ans += a[i] * a[j] * a[(i * j) % 9]
print(ans)
``` | output | 1 | 36,048 | 22 | 72,097 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,049 | 22 | 72,098 |
Tags: number theory
Correct Solution:
```
def main():
ans = 0
a = [0] * 10
n = int(input())
for i in range (1, 9):
a[i] = n // 9 + int(n % 9 >= i)
a[9] = n // 9
for i in range(1, 10):
for j in range(1, 10):
k = i * j % 9
if k == 0:
k = 9
ans += a[i] * a[j] * a[k]
for i in range(1, n + 1):
ans -= n // i
print(ans)
return
if __name__ == "__main__":
main()
``` | output | 1 | 36,049 | 22 | 72,099 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,050 | 22 | 72,100 |
Tags: number theory
Correct Solution:
```
__author__ = 'Darren'
# d(x) = (x-1) % 9 + 1
def solve():
n = int(input())
# count[i]: the number of x's in [1,n] such that d(x) = i
count = [0] * 10
for i in range((n-1) % 9 + 2):
count[i] = (n + 8) // 9
for i in range((n-1) % 9 + 2, 10):
count[i] = n // 9
result = 0
# Count all triples (i, j, k) such that d(d(i)*d(j)) = d(k)
for i in range(1, 10):
for j in range(1, 10):
result += count[i] * count[j] * count[(i*j-1) % 9 + 1]
# For each i, there are n/i triples (i,j,k) such that i*j = k,
# i.e., the correct cases
for i in range(1, n+1):
result -= n // i
print(result)
if __name__ == '__main__':
solve()
``` | output | 1 | 36,050 | 22 | 72,101 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,051 | 22 | 72,102 |
Tags: number theory
Correct Solution:
```
__author__ = 'Darren'
# d(x) = (x-1) % 9 + 1
def solve():
n = int(input())
# count[i]: the number of x's in [1,n] such that d(x) = i
count = [0] * 10
for i in range((n-1) % 9 + 2):
count[i] = (n + 8) // 9
for i in range((n-1) % 9 + 2, 10):
count[i] = n // 9
result = 0
# Count all triples (i, j, k) such that d(d(i)*d(j)) = d(k)
for i in range(1, 10):
for j in range(1, 10):
result += count[i] * count[j] * count[(i*j-1) % 9 + 1]
# For each i, there are n/i triples (i,j,k) such that i*j = k,
# i.e., the correct cases
for i in range(1, n+1):
result -= n // i
print(result)
if __name__ == '__main__':
solve()
# Made By Mostafa_Khaled
``` | output | 1 | 36,051 | 22 | 72,103 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,052 | 22 | 72,104 |
Tags: number theory
Correct Solution:
```
def digital_root(x: int) -> int:
if x <= 9:
return x
tot = 0
while x:
tot += x % 10
x //= 10
return digital_root(tot)
n = int(input())
counts = [0]*10
d = [0]*(n+1)
for i in range(1,n+1):
d[i] = digital_root(i)
counts[d[i]] += 1
answer = 0
# number of triples such that relation holds. i = d(c) j = d(a) k = d(b)
for i in range(1,10):
for j in range(1,10):
for k in range(1,10):
if i == digital_root(j*k):
answer += counts[i]*counts[j]*counts[k]
# subtract number of triples such that the relation holds but c = a*b
for a in range(1,n+1):
for c in range(a,n+1,a):
b = c//a
if d[c] == d[d[a]*d[b]]:
answer -= 1
print(answer)
``` | output | 1 | 36,052 | 22 | 72,105 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3). | instruction | 0 | 36,053 | 22 | 72,106 |
Tags: number theory
Correct Solution:
```
def main():
ans = 0
a = [0] * 10
n = int(input())
for i in range (0, 9):
a[i] = n // 9 + int(n % 9 >= i)
a[0] -= 1
for i in range(0, 9):
for j in range(0, 9):
k = i * j % 9
ans += a[i] * a[j] * a[k]
for i in range(1, n + 1):
ans -= n // i
print(ans)
return
if __name__ == "__main__":
main()
``` | output | 1 | 36,053 | 22 | 72,107 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Not long ago Billy came across such a problem, where there were given three natural numbers A, B and C from the range [1, N], and it was asked to check whether the equation AB = C is correct. Recently Billy studied the concept of a digital root of a number. We should remind you that a digital root d(x) of the number x is the sum s(x) of all the digits of this number, if s(x) β€ 9, otherwise it is d(s(x)). For example, a digital root of the number 6543 is calculated as follows: d(6543) = d(6 + 5 + 4 + 3) = d(18) = 9. Billy has counted that the digital root of a product of numbers is equal to the digital root of the product of the factors' digital roots, i.e. d(xy) = d(d(x)d(y)). And the following solution to the problem came to his mind: to calculate the digital roots and check if this condition is met. However, Billy has doubts that this condition is sufficient. That's why he asks you to find out the amount of test examples for the given problem such that the algorithm proposed by Billy makes mistakes.
Input
The first line contains the only number N (1 β€ N β€ 106).
Output
Output one number β the amount of required A, B and C from the range [1, N].
Examples
Input
4
Output
2
Input
5
Output
6
Note
For the first sample the required triples are (3, 4, 3) and (4, 3, 3).
Submitted Solution:
```
'''
Jana Goodman6
10C Digital Root
this is true: d(xy) = d(d(x)d(y))
need to count AB
question is how many d(AB) == d(C)
but AB != C
'''
import math
import time
def dr(n):
return (n - 1) % 9 + 1
def digsum(n):
return sum(map(int, list(str(n))))
def persist(n):
m = n
ans = 1
while True:
x = digsum(m)
if x < 10:
return ans
ans += 1
m = x
def bad_ones(n):
# take advantage of the repeating nature of dig_root
# and that d(ab) == d(d(a)d(b))
ndiv9, nmod9 = n // 9, n % 9
if nmod9 == 0:
drs = [ndiv9] * 10
else:
drs = [ndiv9 + 1] * (nmod9 + 1) + [ndiv9] * (9 - nmod9)
answer = sum(drs[i] * drs[j] * drs[dr(i * j)]
for j in range(1, 10)
for i in range(1, 10))
answer -= sum(n // i for i in range(1, n + 1))
return answer
if __name__ == '__main__':
print(bad_ones(int(input().strip())))
``` | instruction | 0 | 36,056 | 22 | 72,112 |
Yes | output | 1 | 36,056 | 22 | 72,113 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,410 | 22 | 72,820 |
Tags: brute force, implementation, math
Correct Solution:
```
p, q = map(int,input().split())
n = int(input())
l = list(map(int, input().split()))
for i in l:
if q == 0 or i > p // q:
print('NO')
exit()
p -= i * q
p, q = q, p
print('YES' if q == 0 else 'NO')
``` | output | 1 | 36,410 | 22 | 72,821 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,411 | 22 | 72,822 |
Tags: brute force, implementation, math
Correct Solution:
```
p,q=list(map(int,input().split()))
n=int(input())
ara= list(map(int,input().split()))
b=0
a=1
l=0
for i in ara[::-1]:
l=a
a=(i*a)+b
b=l
#print(a*q,b*p)
if a*q==b*p:
print("YES")
else:
print("NO")
``` | output | 1 | 36,411 | 22 | 72,823 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,412 | 22 | 72,824 |
Tags: brute force, implementation, math
Correct Solution:
```
from decimal import *
getcontext().prec = 75
p = input().split()
n = int(input())
c = input().split()
a='0'
for i in range(n-1,0,-1):
a=str(Decimal(a)+Decimal(c[i]))
a=str(Decimal('1')/Decimal(str(a)))
a=str(Decimal(a)+Decimal(c[0]))
b=str(Decimal(p[0])/Decimal(p[1]))
if(a==b):
print("YES")
else:
print("NO")
``` | output | 1 | 36,412 | 22 | 72,825 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,413 | 22 | 72,826 |
Tags: brute force, implementation, math
Correct Solution:
```
p,q=tuple(map(int,input().split()))
n1=input().split()
n=int(n1[0])
a=list(map(int,input().split()))
x=1
y=a[n-1]
for i in range (n-2,-1,-1):
x1=a[i]*y+x
x=y
y=x1
if p*x==q*y:
print('YES')
else:
print('NO')
``` | output | 1 | 36,413 | 22 | 72,827 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,414 | 22 | 72,828 |
Tags: brute force, implementation, math
Correct Solution:
```
P,Q=map(int,input().split())
n=int(input())
a = [int(i) for i in input().split()]
p=[]
q=[]
p.append(a[0])
q.append(1)
if (n==1):
if(p[0]*Q==q[0]*P):
print("YES")
else:
print("NO")
else:
p.append(a[1]*a[0]+1)
q.append(a[1])
for i in range(2,n):
p.append(a[i]*p[i-1]+p[i-2])
q.append(a[i]*q[i-1]+q[i-2])
if (p[n-1]*Q==q[n-1]*P):
print("YES")
else:
print("NO")
``` | output | 1 | 36,414 | 22 | 72,829 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,415 | 22 | 72,830 |
Tags: brute force, implementation, math
Correct Solution:
```
from fractions import Fraction
pq = Fraction(*map(int, input().split()))
input()
a = list(map(int, input().split()))
a.reverse()
f = Fraction(a[0], 1)
for i in a[1:]:
f = i + Fraction(1, f)
if pq == f:
print("YES")
else:
print("NO")
``` | output | 1 | 36,415 | 22 | 72,831 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,416 | 22 | 72,832 |
Tags: brute force, implementation, math
Correct Solution:
```
p, q = map(int, input().split())
n = int(input())
A = list(map(int, input().split()))
for i in range(n):
p, q = q, p - A[i] * q
if p != 0 and q == 0:
print('YES')
else:
print('NO')
# Made By Mostafa_Khaled
``` | output | 1 | 36,416 | 22 | 72,833 |
Provide tags and a correct Python 3 solution for this coding contest problem.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>. | instruction | 0 | 36,417 | 22 | 72,834 |
Tags: brute force, implementation, math
Correct Solution:
```
def gcd(a,b):
if(b==0):
return a
return gcd(b,a%b)
p,q=map(int,input().split())
N=int(input())
A=list(map(int,input().split()))
znam=1
chisl=A[N-1]
for i in range (N-1,0,-1):
chisl,znam=znam,chisl
#print(chisl,znam)
chisl += znam*A[i-1]
#print(chisl,znam)
if(p*znam==q*chisl):
print('YES')
else:
print('NO')
``` | output | 1 | 36,417 | 22 | 72,835 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>.
Submitted Solution:
```
arr = [0 for i in range(100)]
def gcd(a,b):
if(b == 0): return a
return gcd(b,a%b)
def compute(a,b,c):
ret = [a*c[0]+b*c[1],c[0]]
return tuple(ret)
def dfs(idx):
if(idx == n-2): return compute(arr[idx],1,tuple([arr[idx+1],1]))
return compute(arr[idx],1,dfs(idx+1))
p,q = list(map(int,input().split()))
n = int(input())
arr = list(map(int,input().split()))
if(n >= 2): ans = dfs(0)
else: ans = tuple([arr[0],1])
d = gcd(ans[0],ans[1])
ans = tuple([ans[0]//d,ans[1]//d])
d = gcd(p,q)
ans2 = tuple([p//d,q//d])
if(ans[0] == ans2[0] and ans[1] == ans2[1]):
print("YES")
else: print("NO")
``` | instruction | 0 | 36,418 | 22 | 72,836 |
Yes | output | 1 | 36,418 | 22 | 72,837 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
A continued fraction of height n is a fraction of form <image>. You are given two rational numbers, one is represented as <image> and the other one is represented as a finite fraction of height n. Check if they are equal.
Input
The first line contains two space-separated integers p, q (1 β€ q β€ p β€ 1018) β the numerator and the denominator of the first fraction.
The second line contains integer n (1 β€ n β€ 90) β the height of the second fraction. The third line contains n space-separated integers a1, a2, ..., an (1 β€ ai β€ 1018) β the continued fraction.
Please, do not use the %lld specifier to read or write 64-bit integers in Π‘++. It is preferred to use the cin, cout streams or the %I64d specifier.
Output
Print "YES" if these fractions are equal and "NO" otherwise.
Examples
Input
9 4
2
2 4
Output
YES
Input
9 4
3
2 3 1
Output
YES
Input
9 4
3
1 2 4
Output
NO
Note
In the first sample <image>.
In the second sample <image>.
In the third sample <image>.
Submitted Solution:
```
from math import gcd
q,p = map(int,input().split())
n = int(input())
t = [*map(int,input().split())]
d = gcd(p,q)
p//=d
q//=d
a=int(1)
b=int(t[n-1])
i=n-2
while i>=0 :
a=b*t[i]+a
d = gcd(a,b)
a//=d
b//=d
i-=1
c=a
a=b
b=c
if a==p and q==b :
print("YES")
else :
print("NO")
``` | instruction | 0 | 36,419 | 22 | 72,838 |
Yes | output | 1 | 36,419 | 22 | 72,839 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Stepan has a very big positive integer.
Let's consider all cyclic shifts of Stepan's integer (if we look at his integer like at a string) which are also integers (i.e. they do not have leading zeros). Let's call such shifts as good shifts. For example, for the integer 10203 the good shifts are the integer itself 10203 and integers 20310 and 31020.
Stepan wants to know the minimum remainder of the division by the given number m among all good shifts. Your task is to determine the minimum remainder of the division by m.
Input
The first line contains the integer which Stepan has. The length of Stepan's integer is between 2 and 200 000 digits, inclusive. It is guaranteed that Stepan's integer does not contain leading zeros.
The second line contains the integer m (2 β€ m β€ 108) β the number by which Stepan divides good shifts of his integer.
Output
Print the minimum remainder which Stepan can get if he divides all good shifts of his integer by the given number m.
Examples
Input
521
3
Output
2
Input
1001
5
Output
0
Input
5678901234567890123456789
10000
Output
123
Note
In the first example all good shifts of the integer 521 (good shifts are equal to 521, 215 and 152) has same remainder 2 when dividing by 3.
In the second example there are only two good shifts: the Stepan's integer itself and the shift by one position to the right. The integer itself is 1001 and the remainder after dividing it by 5 equals 1. The shift by one position to the right equals to 1100 and the remainder after dividing it by 5 equals 0, which is the minimum possible remainder. | instruction | 0 | 36,569 | 22 | 73,138 |
Tags: *special, math, number theory
Correct Solution:
```
s = input()
m = int(input())
pw = [0] * (len(s) + 1)
pw[0] = 1
for i in range(1, len(s) + 1):
pw[i] = pw[i - 1] * 10 % m
cur = 0
for i in range(len(s)):
cur *= 10
cur += ord(s[i]) - ord('0')
cur %= m
ans = cur
for i in range(1, len(s)):
cur *= 10
cur %= m
cur -= ((ord(s[i - 1]) - ord('0')) * pw[len(s)] % m);
cur += m
cur %= m
cur += ord(s[i - 1]) - ord('0')
cur %= m
if (s[i] != '0'):
ans = min(ans, cur)
print(ans)
``` | output | 1 | 36,569 | 22 | 73,139 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Stepan has a very big positive integer.
Let's consider all cyclic shifts of Stepan's integer (if we look at his integer like at a string) which are also integers (i.e. they do not have leading zeros). Let's call such shifts as good shifts. For example, for the integer 10203 the good shifts are the integer itself 10203 and integers 20310 and 31020.
Stepan wants to know the minimum remainder of the division by the given number m among all good shifts. Your task is to determine the minimum remainder of the division by m.
Input
The first line contains the integer which Stepan has. The length of Stepan's integer is between 2 and 200 000 digits, inclusive. It is guaranteed that Stepan's integer does not contain leading zeros.
The second line contains the integer m (2 β€ m β€ 108) β the number by which Stepan divides good shifts of his integer.
Output
Print the minimum remainder which Stepan can get if he divides all good shifts of his integer by the given number m.
Examples
Input
521
3
Output
2
Input
1001
5
Output
0
Input
5678901234567890123456789
10000
Output
123
Note
In the first example all good shifts of the integer 521 (good shifts are equal to 521, 215 and 152) has same remainder 2 when dividing by 3.
In the second example there are only two good shifts: the Stepan's integer itself and the shift by one position to the right. The integer itself is 1001 and the remainder after dividing it by 5 equals 1. The shift by one position to the right equals to 1100 and the remainder after dividing it by 5 equals 0, which is the minimum possible remainder. | instruction | 0 | 36,570 | 22 | 73,140 |
Tags: *special, math, number theory
Correct Solution:
```
s=input()
m=int(input())
N=int(s)
MIN=N%m
L=len(s)-1
temp1=N
temp2=0
big=pow(10,L,m)
for i in range(0,len(s)-1):
temp2*=10
temp2+=int(s[i])
temp2%=m
temp1-=big*int(s[i])
temp1*=10
temp1%=m
temp=temp1+temp2
#print(temp1,temp2)
#print(temp)
temp%=m
if s[i+1]!='0':
if temp<MIN:
MIN=temp
print(MIN)
``` | output | 1 | 36,570 | 22 | 73,141 |
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