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Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,807 | 22 | 107,614 |
"Correct Solution:
```
import sys
a=[0,0,1]+[1,0]*499999
for i in range(3,999,2):
if a[i]:a[i*i::i]=[0]*len(a[i*i::i])
for e in sys.stdin:print(sum(a[:int(e)+1]))
``` | output | 1 | 53,807 | 22 | 107,615 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,808 | 22 | 107,616 |
"Correct Solution:
```
import sys
a=[0,0]+[1]*10**6
for i in range(999):
if a[i]:a[i*2::i]=[0 for j in a[i*2::i]]
for e in sys.stdin:print(sum(a[:int(e)+1]))
``` | output | 1 | 53,808 | 22 | 107,617 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,809 | 22 | 107,618 |
"Correct Solution:
```
import sys
m=166666;s=[1]*m;t=[1]*m
for i in range(m):
if s[i]:
k=6*i+5
s[i+k::k]=[0]*len(s[i+k::k])
t[-2-i+k::k]=[0]*len(t[-2-i+k::k])
if t[i]:
k=6*i+7
s[-2-i+k::k]=[0]*len(s[-2-i+k::k])
t[i+k::k]=[0]*len(t[i+k::k])
for e in map(int,sys.stdin):
print([e-1,sum(s[:(e+1)//6])+sum(t[:(e-1)//6])+2][e>3])
``` | output | 1 | 53,809 | 22 | 107,619 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,810 | 22 | 107,620 |
"Correct Solution:
```
import bisect
def ifprime(n):
for i in range(3,int(n**0.5)+1):
if n%i==0:return(False)
return(True)
prime_list=[2,3]
while 1:
try:
n=int(input())
if n>prime_list[-1]+1:
for i in range(prime_list[-1]+2,n+1,2):
if ifprime(i):prime_list.append(i)
x=bisect.bisect_left(prime_list,n+1)
print(x)
except:break
``` | output | 1 | 53,810 | 22 | 107,621 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,811 | 22 | 107,622 |
"Correct Solution:
```
#79 素数の数
import math
while(1):
try:
n = int(input())
except:
break
m = int(math.sqrt(n))
keys = [n // i for i in range(1, m+1)] #1からm+1の間のみn//iを行う
keys += range(keys[-1]-1, 0, -1) #range(始まりの数値,最後の数値,増加量)
h = {i: i-1 for i in keys}
for i in range(2, m+1):
if h [i] > h[i-1]:
hp = h[i-1]
i2 = i*i
for j in keys:
if j < i2:
break
h[j] -= h[j // i] - hp
print(h[n])
``` | output | 1 | 53,811 | 22 | 107,623 |
Provide a correct Python 3 solution for this coding contest problem.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5 | instruction | 0 | 53,812 | 22 | 107,624 |
"Correct Solution:
```
import sys
m=500000;a=[1]*m
for i in range(3,999,2):
if a[i//2]:a[i*i//2::i]=[0]*((m-1-i*i//2)//i+1)
for e in map(int,sys.stdin):print([e-1,sum(a[:(e+1)//2])][e>3])
``` | output | 1 | 53,812 | 22 | 107,625 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
#!/usr/bin/env python3
import itertools
import math
import sys
MAX = 999999
def sieve_of_eratosthenes(max):
is_prime = [True] * (max + 1)
is_prime[0] = False
is_prime[1] = False
for i in range(2, int(math.sqrt(max)) + 1):
if not is_prime[i]:
continue
for j in range(i * i, max + 1, i):
is_prime[j] = False
return filter(lambda x: is_prime[x], range(max + 1))
def main():
p = list(sieve_of_eratosthenes(MAX))
for l in sys.stdin:
n = int(l)
print(len(list(itertools.takewhile(lambda x: x <= n, p))))
if __name__ == '__main__':
main()
``` | instruction | 0 | 53,813 | 22 | 107,626 |
Yes | output | 1 | 53,813 | 22 | 107,627 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
import sys
def prime(n):
a = [0] * (n + 1)
a[0] = 1
a[1] = 1
i = 2
while n // i >= i:
if a[i] != 0:
i += 1
continue
j = 2 * i
while j <= n:
a[j] = 1
j += i
i += 1
return [i for i, ai in enumerate(a) if ai == 0]
for i in sys.stdin:
print(len(prime(int(i))))
``` | instruction | 0 | 53,814 | 22 | 107,628 |
Yes | output | 1 | 53,814 | 22 | 107,629 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
def primes(n):
is_prime = [True] * (n + 1)
is_prime[0] = False
is_prime[1] = False
for i in range(2, int(n**0.5) + 1):
if not is_prime[i]:
continue
for j in range(i * 2, n + 1, i):
is_prime[j] = False
return [i for i in range(n + 1) if is_prime[i]]
if __name__ == "__main__":
while True:
try:
n = int(input())
print(len(primes(n)))
except:
break
``` | instruction | 0 | 53,815 | 22 | 107,630 |
Yes | output | 1 | 53,815 | 22 | 107,631 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
plst = [1 for i in range(1000001)]
plst[0] = plst[1] = 0
for i in range(2,1000001):
if plst[i] == 1:
for j in range(i * 2,1000001,i):
plst[j] = 0
while True:
try:
print(sum(plst[:int(input()) + 1]))
except EOFError:
break
``` | instruction | 0 | 53,816 | 22 | 107,632 |
Yes | output | 1 | 53,816 | 22 | 107,633 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
import sys
def primes(n):
f = []
for i in range(n + 1):
if i > 2 and i % 2 == 0:
f.append(0)
else:
f.append(1)
i = 3
while i * i <= n:
if f[i] == 1:
j = i * i
while j <= n:
f[j] = 0
j += i + i
i += 2
return f[2:]
if __name__ == "__main__":
ps = primes(100)
for n in sys.stdin:
i = int(n)
if i < 2:
print(i, 0)
else:
print(i, sum(ps[:i - 1]))
``` | instruction | 0 | 53,817 | 22 | 107,634 |
No | output | 1 | 53,817 | 22 | 107,635 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
import
def prime_calc(n):
if n < 2:
return False
else:
i = 2
while n > i:
if n % i == 0:
return False
else:
i += 1
return True
def prime(n):
cnt = 0
for i in range(0, n+1):
ans = prime_calc(i)
if ans is True:
cnt = cnt + 1
return cnt
def main():
a = []
for line in sys.stdin:
a.append(int(line))
for line in a:
print(prime(line))
if __name__ == "__main__":
main()
``` | instruction | 0 | 53,818 | 22 | 107,636 |
No | output | 1 | 53,818 | 22 | 107,637 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
import math
def AtkinSieve (limit):
results = [2,3,5]
sieve = [False]*(limit+1)
factor = int(math.sqrt(limit))+1
for i in range(1,factor):
for j in range(1, factor):
n = 4*i**2+j**2
if (n <= limit) and (n % 12 == 1 or n % 12 == 5):
sieve[n] = not sieve[n]
n = 3*i**2+j**2
if (n <= limit) and (n % 12 == 7):
sieve[n] = not sieve[n]
if i>j:
n = 3*i**2-j**2
if (n <= limit) and (n % 12 == 11):
sieve[n] = not sieve[n]
for index in range(5,factor):
if sieve[index]:
for jndex in range(index**2, limit, index**2):
sieve[jndex] = False
for index in range(7,limit):
if sieve[index]:
results.append(index)
return results
while(True):
try:
n = input()
except EOFError:
break
n = int(n)
print(len(AtkinSieve(n)))
``` | instruction | 0 | 53,819 | 22 | 107,638 |
No | output | 1 | 53,819 | 22 | 107,639 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Write a program which reads an integer n and prints the number of prime numbers which are less than or equal to n. A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the first four prime numbers are: 2, 3, 5 and 7.
Input
Input consists of several datasets. Each dataset has an integer n (1 ≤ n ≤ 999,999) in a line.
The number of datasets is less than or equal to 30.
Output
For each dataset, prints the number of prime numbers.
Example
Input
10
3
11
Output
4
2
5
Submitted Solution:
```
import sys
def fact(n):
ret = [1]
for i in range(2,int(n//2)+1):
if n%i==0:
ret.append(i)
return ret
def isPrime(n):
if n<=1:
return False
if n==2:
return True
if fact(n)==[1]:
return True
else:
return False
lines = sys.stdin.readlines()
for line in lines:
inp = int(line)
s=0
for i in range(2,inp+1):
if isPrime(i):
s+=1
print (s)
``` | instruction | 0 | 53,820 | 22 | 107,640 |
No | output | 1 | 53,820 | 22 | 107,641 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,035 | 22 | 108,070 |
Tags: brute force, implementation
Correct Solution:
```
from collections import Counter
l, r = [int(n) for n in input().split()]
def find(l, r):
for i in range(l, r+1):
if Counter(str(i)).most_common(1)[0][1] == 1:
return i
return -1
print(find(l,r))
``` | output | 1 | 54,035 | 22 | 108,071 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,036 | 22 | 108,072 |
Tags: brute force, implementation
Correct Solution:
```
def solve(n):
d={}
l=[]
while(n>0):
l.append(n%10)
n=n//10
for t in l:
val=d.get(t,0)
d[t]=val+1
if d[t]>1:
return False
return True
l,r=map(int,input().split())
flag=0
for t in range(l,r+1):
k=solve(t)
if(k):
print(t)
flag=1
break
if(not flag):
print(-1)
``` | output | 1 | 54,036 | 22 | 108,073 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,037 | 22 | 108,074 |
Tags: brute force, implementation
Correct Solution:
```
l, r = [int(x) for x in input().split(' ')]
for i in range(l, r + 1):
if len(str(i)) == len(set(str(i))):
print(i)
break
else:
print(-1)
``` | output | 1 | 54,037 | 22 | 108,075 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,039 | 22 | 108,078 |
Tags: brute force, implementation
Correct Solution:
```
def isUniqueDigit(s):
for d in ['0','1','2','3','4','5','6','7','8','9']:
if s.count(d) > 1:
return False
return True
s = input().split()
l, r = int(s[0]), int(s[1])
found = '-1'
for i in range(l, r+1):
if isUniqueDigit(str(i)):
found = str(i)
break
print(found)
``` | output | 1 | 54,039 | 22 | 108,079 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,040 | 22 | 108,080 |
Tags: brute force, implementation
Correct Solution:
```
l,r=map(int,input().split())
a=-1
for i in range(l,r+1):
b=str(i)
if len(b)==len(set(b)):a=i;break
print(a)
``` | output | 1 | 54,040 | 22 | 108,081 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You have two integers l and r. Find an integer x which satisfies the conditions below:
* l ≤ x ≤ r.
* All digits of x are different.
If there are multiple answers, print any of them.
Input
The first line contains two integers l and r (1 ≤ l ≤ r ≤ 10^{5}).
Output
If an answer exists, print any of them. Otherwise, print -1.
Examples
Input
121 130
Output
123
Input
98766 100000
Output
-1
Note
In the first example, 123 is one of the possible answers. However, 121 can't be the answer, because there are multiple 1s on different digits.
In the second example, there is no valid answer. | instruction | 0 | 54,041 | 22 | 108,082 |
Tags: brute force, implementation
Correct Solution:
```
a,b=list(map(int,input().split(' ')))
for i in range(a,b+1):
s=str(i)
x=-1
if len(s)==len(set(s)):
x=i
break
print(x)
``` | output | 1 | 54,041 | 22 | 108,083 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on a math lesson the teacher told Vovochka that the Euler function of a positive integer φ(n) is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The number 1 is coprime to all the positive integers and φ(1) = 1.
Now the teacher gave Vovochka an array of n positive integers a1, a2, ..., an and a task to process q queries li ri — to calculate and print <image> modulo 109 + 7. As it is too hard for a second grade school student, you've decided to help Vovochka.
Input
The first line of the input contains number n (1 ≤ n ≤ 200 000) — the length of the array given to Vovochka. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 106).
The third line contains integer q (1 ≤ q ≤ 200 000) — the number of queries. Next q lines contain the queries, one per line. Each query is defined by the boundaries of the segment li and ri (1 ≤ li ≤ ri ≤ n).
Output
Print q numbers — the value of the Euler function for each query, calculated modulo 109 + 7.
Examples
Input
10
1 2 3 4 5 6 7 8 9 10
7
1 1
3 8
5 6
4 8
8 10
7 9
7 10
Output
1
4608
8
1536
192
144
1152
Input
7
24 63 13 52 6 10 1
6
3 5
4 7
1 7
2 4
3 6
2 6
Output
1248
768
12939264
11232
9984
539136
Note
In the second sample the values are calculated like that:
* φ(13·52·6) = φ(4056) = 1248
* φ(52·6·10·1) = φ(3120) = 768
* φ(24·63·13·52·6·10·1) = φ(61326720) = 12939264
* φ(63·13·52) = φ(42588) = 11232
* φ(13·52·6·10) = φ(40560) = 9984
* φ(63·13·52·6·10) = φ(2555280) = 539136
Submitted Solution:
```
from sys import stdin,stdout
import math
mod=10**9 + 7
def build(a,segtree,curr,low,end):
if low==end:
segtree[curr]=a[end]
return a[end]
mid=low+(end-low)//2
y=build(a,segtree,2*curr+1,low,mid)
y1=build(a,segtree,2*curr+2,mid+1,end)
segtree[curr]=(y*y1)%mod
return segtree[curr]%mod
def query(a,segtree,curr,low,end,index1,index2):
if end<index1 or low>index2:
return 1
if low>=index1 and end<=index2:
return segtree[curr]
mid=low+(end-low)//2
y=query(a,segtree,2*curr+1,low,mid,index1,index2)
y1=query(a,segtree,2*curr+2,mid+1,end,index1,index2)
return (y*y1)%mod
etf=[i for i in range(10**6 + 1)]
for i in range(2,10**6+1):
if etf[i]==i:
etf[i]=i-1
for j in range(2*i,10**6+1,i):
etf[j]-=etf[j]//i
etf[1]=1
etf[0]=0
n=int(stdin.readline())
a=list(map(int,stdin.readline().split()))
b=[0]*(n)
for i in range(n):
b[i]=etf[a[i]]
print(b)
segtree=[1]*(2*(2**(math.ceil(math.log(n,2))))-1)
a=[]+b
build(a,segtree,0,0,n-1)
print(segtree)
q=int(input())
for i in range(q):
l,r=list(map(int,stdin.readline().split()))
print(query(a,segtree,0,0,n-1,l-1,r-1))
``` | instruction | 0 | 54,388 | 22 | 108,776 |
No | output | 1 | 54,388 | 22 | 108,777 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on a math lesson the teacher told Vovochka that the Euler function of a positive integer φ(n) is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The number 1 is coprime to all the positive integers and φ(1) = 1.
Now the teacher gave Vovochka an array of n positive integers a1, a2, ..., an and a task to process q queries li ri — to calculate and print <image> modulo 109 + 7. As it is too hard for a second grade school student, you've decided to help Vovochka.
Input
The first line of the input contains number n (1 ≤ n ≤ 200 000) — the length of the array given to Vovochka. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 106).
The third line contains integer q (1 ≤ q ≤ 200 000) — the number of queries. Next q lines contain the queries, one per line. Each query is defined by the boundaries of the segment li and ri (1 ≤ li ≤ ri ≤ n).
Output
Print q numbers — the value of the Euler function for each query, calculated modulo 109 + 7.
Examples
Input
10
1 2 3 4 5 6 7 8 9 10
7
1 1
3 8
5 6
4 8
8 10
7 9
7 10
Output
1
4608
8
1536
192
144
1152
Input
7
24 63 13 52 6 10 1
6
3 5
4 7
1 7
2 4
3 6
2 6
Output
1248
768
12939264
11232
9984
539136
Note
In the second sample the values are calculated like that:
* φ(13·52·6) = φ(4056) = 1248
* φ(52·6·10·1) = φ(3120) = 768
* φ(24·63·13·52·6·10·1) = φ(61326720) = 12939264
* φ(63·13·52) = φ(42588) = 11232
* φ(13·52·6·10) = φ(40560) = 9984
* φ(63·13·52·6·10) = φ(2555280) = 539136
Submitted Solution:
```
'''
Range euler queries
'''
from math import ceil, log2
mod = 10**9+7
def totient(n):
res = n
p = 2
while p*p <= n:
if n%p == 0:
while n%p == 0:
n//=p
res = (res%mod * (1.0 - 1.0/float(p))%mod)%mod
p+=1
if n > 1:
res = (res%mod * (1.0 - 1.0/float(n))%mod)%mod
return int(res)
def get_query(st, x, y, start, end, si):
if start >= x and end <= y:
return st[si]
if start > y or end < x:
return 1
mid = (start + end) >> 1
left = get_query(st, x, y, start, mid, 2*si+1)
right = get_query(st, x, y, mid+1, end, 2*si+2)
return (left%mod * right%mod)%mod
def constructST(arr, st, start, end, si):
if start == end:
st[si] = arr[start]
return st[si]
mid = (start+end) >> 1
left = constructST(arr, st, start, mid, 2*si+1)
right = constructST(arr, st, mid+1, end, 2*si+2)
st[si] = (left%mod * right%mod)%mod
return st[si]
n = int(input())
arr = [int(i) for i in input().split()]
q = int(input())
height = ceil(log2(n))
size = pow(2, height+1) - 1
st = [1 for i in range(size)]
constructST(arr, st, 0, n-1, 0)
# print(st)
while q > 0:
l, r = map(int, input().split())
res = get_query(st, l-1, r-1, 0, n-1, 0)
# print(res)
print(totient(res))
q-=1
``` | instruction | 0 | 54,389 | 22 | 108,778 |
No | output | 1 | 54,389 | 22 | 108,779 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on a math lesson the teacher told Vovochka that the Euler function of a positive integer φ(n) is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The number 1 is coprime to all the positive integers and φ(1) = 1.
Now the teacher gave Vovochka an array of n positive integers a1, a2, ..., an and a task to process q queries li ri — to calculate and print <image> modulo 109 + 7. As it is too hard for a second grade school student, you've decided to help Vovochka.
Input
The first line of the input contains number n (1 ≤ n ≤ 200 000) — the length of the array given to Vovochka. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 106).
The third line contains integer q (1 ≤ q ≤ 200 000) — the number of queries. Next q lines contain the queries, one per line. Each query is defined by the boundaries of the segment li and ri (1 ≤ li ≤ ri ≤ n).
Output
Print q numbers — the value of the Euler function for each query, calculated modulo 109 + 7.
Examples
Input
10
1 2 3 4 5 6 7 8 9 10
7
1 1
3 8
5 6
4 8
8 10
7 9
7 10
Output
1
4608
8
1536
192
144
1152
Input
7
24 63 13 52 6 10 1
6
3 5
4 7
1 7
2 4
3 6
2 6
Output
1248
768
12939264
11232
9984
539136
Note
In the second sample the values are calculated like that:
* φ(13·52·6) = φ(4056) = 1248
* φ(52·6·10·1) = φ(3120) = 768
* φ(24·63·13·52·6·10·1) = φ(61326720) = 12939264
* φ(63·13·52) = φ(42588) = 11232
* φ(13·52·6·10) = φ(40560) = 9984
* φ(63·13·52·6·10) = φ(2555280) = 539136
Submitted Solution:
```
'''
Range euler queries
'''
from math import ceil, log2
mod = 10**9+7
def totient(n):
res = n
p = 2
while p*p <= n:
if n%p == 0:
while n%p == 0:
n//=p
res = res * (1.0 - 1.0/float(p))
p+=1
if n > 1:
res = res * (1.0 - 1.0/float(n))
return int(res)
def get_query(st, x, y, start, end, si):
if start >= x and end <= y:
return st[si]
if start > y or end < x:
return 1
mid = (start + end) >> 1
left = get_query(st, x, y, start, mid, 2*si+1)
right = get_query(st, x, y, mid+1, end, 2*si+2)
return (left%mod * right%mod)%mod
def constructST(arr, st, start, end, si):
if start == end:
st[si] = arr[start]
return st[si]
mid = (start+end) >> 1
left = constructST(arr, st, start, mid, 2*si+1)
right = constructST(arr, st, mid+1, end, 2*si+2)
st[si] = (left%mod * right%mod)%mod
return st[si]
n = int(input())
arr = [int(i) for i in input().split()]
q = int(input())
height = ceil(log2(n))
size = pow(2, height+1) - 1
st = [1 for i in range(size)]
constructST(arr, st, 0, n-1, 0)
# print(st)
while q > 0:
l, r = map(int, input().split())
res = get_query(st, l-1, r-1, 0, n-1, 0)
# print(res)
print(totient(res))
q-=1
``` | instruction | 0 | 54,390 | 22 | 108,780 |
No | output | 1 | 54,390 | 22 | 108,781 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Today on a math lesson the teacher told Vovochka that the Euler function of a positive integer φ(n) is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The number 1 is coprime to all the positive integers and φ(1) = 1.
Now the teacher gave Vovochka an array of n positive integers a1, a2, ..., an and a task to process q queries li ri — to calculate and print <image> modulo 109 + 7. As it is too hard for a second grade school student, you've decided to help Vovochka.
Input
The first line of the input contains number n (1 ≤ n ≤ 200 000) — the length of the array given to Vovochka. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 106).
The third line contains integer q (1 ≤ q ≤ 200 000) — the number of queries. Next q lines contain the queries, one per line. Each query is defined by the boundaries of the segment li and ri (1 ≤ li ≤ ri ≤ n).
Output
Print q numbers — the value of the Euler function for each query, calculated modulo 109 + 7.
Examples
Input
10
1 2 3 4 5 6 7 8 9 10
7
1 1
3 8
5 6
4 8
8 10
7 9
7 10
Output
1
4608
8
1536
192
144
1152
Input
7
24 63 13 52 6 10 1
6
3 5
4 7
1 7
2 4
3 6
2 6
Output
1248
768
12939264
11232
9984
539136
Note
In the second sample the values are calculated like that:
* φ(13·52·6) = φ(4056) = 1248
* φ(52·6·10·1) = φ(3120) = 768
* φ(24·63·13·52·6·10·1) = φ(61326720) = 12939264
* φ(63·13·52) = φ(42588) = 11232
* φ(13·52·6·10) = φ(40560) = 9984
* φ(63·13·52·6·10) = φ(2555280) = 539136
Submitted Solution:
```
'''
Range euler queries
'''
from math import ceil, log2
mod = 10**9+7
def totient(n):
res = n
p = 2
while p*p <= n:
if n%p == 0:
while n%p == 0:
n//=p
res = res * (1.0 - 1.0/float(p))
p+=1
if n > 1:
res = res * (1.0 - 1.0/float(n))
return int(res)
def get_query(st, x, y, start, end, si):
if start >= x and end <= y:
return st[si]
if start > y or end < x:
return 1
mid = (start + end) >> 1
left = get_query(st, x, y, start, mid, 2*si+1)
right = get_query(st, x, y, mid+1, end, 2*si+2)
return (left%mod * right%mod)%mod
def constructST(arr, st, start, end, si):
if start == end:
st[si] = arr[start]
return st[si]
mid = (start+end) >> 1
left = constructST(arr, st, start, mid, 2*si+1)
right = constructST(arr, st, mid+1, end, 2*si+2)
st[si] = (left%mod * right%mod)%mod
return st[si]
n = int(input())
arr = [int(i) for i in input().split()]
q = int(input())
height = ceil(log2(n))
size = pow(2, height+1) - 1
st = [1 for i in range(size)]
constructST(arr, st, 0, n-1, 0)
# print(st)
while q > 0:
l, r = map(int, input().split())
res = get_query(st, l-1, r-1, 0, n-1, 0)
# print(res)
print(totient(res)%mod)
q-=1
``` | instruction | 0 | 54,391 | 22 | 108,782 |
No | output | 1 | 54,391 | 22 | 108,783 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,408 | 22 | 108,816 |
Tags: dp, math
Correct Solution:
```
import sys
def add_zeros(maxs, mins):
diff = len(maxs) - len(mins)
s0 = diff * "0"
return s0 + mins
def result(_xor, _sum, flag):
if len(_xor) + len(_sum) == 0:
return 0 if flag else 1
s = _xor[-1] + _sum[-1]
newxor = _xor[:-1]
newsum = _sum[:-1]
if s == '00':
return 0 if flag else result(newxor, newsum, True) + result(newxor, newsum, False)
if s == '11':
return 0 if flag else 2 * result(newxor, newsum, False)
if s == '10':
return 2 * result(newxor, newsum, True) if flag else 0
if s == '01':
return result(newxor, newsum, True) + result(newxor, newsum, False) if flag else 0
ss, xr = [int(x) for x in sys.stdin.readline().split()]
_sum = bin(ss).split('b')[1]
_xor = bin(xr).split('b')[1]
if len(_sum) > len(_xor):
_xor = add_zeros(_sum, _xor)
else:
_sum = add_zeros(_xor, _sum)
#print(_sum)
#print(_xor)
res = result(_xor, _sum, False)
print(res - 2 if ss == xr else res)
``` | output | 1 | 54,408 | 22 | 108,817 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,409 | 22 | 108,818 |
Tags: dp, math
Correct Solution:
```
s, x = map(int, input().split())
def check(x, k):
while x:
if x & 1 == 1 and k & 1 == 1:
return False
x //= 2
k //= 2
return True
if (s - x) % 2 == 1:
print(0)
else:
k = (s - x) // 2
if not check(x, k):
print(0)
else:
x0 = x & (~k)
ans = 0
while x0:
ans += x0 % 2
x0 //= 2
ans = 2 ** ans
if k == 0:
ans -= 2
print(ans)
``` | output | 1 | 54,409 | 22 | 108,819 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,410 | 22 | 108,820 |
Tags: dp, math
Correct Solution:
```
#!/usr/bin/env python3
from math import log
s, x = [int(x) for x in input().split()]
# sum == 0, xor == 0, one == 0 - both are 0 or 1 (2 variants)
# sum == 0, xor == 0, one == 1 - impossible
# sum == 0, xor == 1, one == 0 - impossible
# sum == 0, xor == 1, one == 1 - one 1 and another 1 is 0 (2 variants)
# sum == 1, xor == 0, one == 0 - impossible
# sum == 1, xor == 0, one == 1 - both are 0 or 1 (2 variants)
# sum == 1, xor == 1, one == 0 - one 1 and another 1 is 0 (2 variants)
# sum == 1, xor == 1, one == 1 - impossible
def get_count(size, one_bit, s, x):
if size == 0 or \
not one_bit and s == 0 and x == 0:
return 1
sum_bit = s & 1 != 0
xor_bit = x & 1 != 0
# print('size = {2:0>2}, sum_bit = {0}, xor_bit = {1}, one_bit = {3}'.format(sum_bit, xor_bit, size, one_bit))
if not sum_bit and not xor_bit and one_bit or \
not sum_bit and xor_bit and not one_bit or \
sum_bit and not xor_bit and not one_bit or \
sum_bit and xor_bit and one_bit:
return 0
s >>= 1
x >>= 1
size -= 1
if not sum_bit and not xor_bit and not one_bit or \
sum_bit and not xor_bit and one_bit:
return get_count(size, False, s, x) + get_count(size, True, s, x)
elif not sum_bit and xor_bit and one_bit:
return 2 * get_count(size, True, s, x)
else:
return 2 * get_count(size, False, s, x)
size = int(log(1000000000000)/log(2)) + 5
count = get_count(size, False, s, x)
if s == x:
assert count >= 2
count -= 2 # numbers have to be > 0, so pairs with 0 aren't suitable
print(count)
``` | output | 1 | 54,410 | 22 | 108,821 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,411 | 22 | 108,822 |
Tags: dp, math
Correct Solution:
```
'''
(a+b)=(a^b)+2*(a&b)
'''
def check(a,b):
# a and vala
for i in range(60):
if a&(1<<i):
if b&(1<<i):
return False
return True
# print(check(1,5))
def f(s,x):
if (s-x)%2!=0:
return 0
nd=(s-x)//2
bd=0
if nd==0:
bd=2
if check(nd,x):
btcnt=(bin(x).count("1"))
return (2**(btcnt))-bd
return 0
a,b=map(int,input().strip().split())
print(f(a,b))
``` | output | 1 | 54,411 | 22 | 108,823 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,412 | 22 | 108,824 |
Tags: dp, math
Correct Solution:
```
import sys
def read_ints():
return map(int, input().split())
s, x = read_ints()
if s < x or ((s - x) & 1):
print(0)
else:
ans = 1
ab = (s - x) // 2
xb = bin(x)[2:][::-1]
abb = bin(ab)[2:][::-1]
abb = abb + '0' * max(0, len(xb) - len(abb))
xb = xb + '0' * max(0, len(abb) - len(xb))
for abi, xbi in zip(abb, xb):
if xbi == '1':
if abi == '0':
ans = (ans << 1)
else:
print(0)
sys.exit(0)
if s == x:
ans -= 2
print(ans)
``` | output | 1 | 54,412 | 22 | 108,825 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,413 | 22 | 108,826 |
Tags: dp, math
Correct Solution:
```
s_, x_ = map(int, input().split())
s = bin(s_)[2:]
x = bin(x_)[2:]
if len(s) < len(x):
print(0)
exit()
x = '0' * (len(s) - len(x)) + x
#print(s)
#print(x)
zero = 1
one = 0
for a, b in zip(x[::-1], s[::-1]):
#print('a, b = ', a, b)
a = int(a)
b = int(b)
new_one = 0
new_zero = 0
if (a == 0 and b == 0):
new_one += zero
new_zero += zero
if (a == 0 and b == 1):
new_zero += one
new_one += one
if (a == 1 and b == 0):
new_one = 2 * one
new_zero = 0
if (a == 1 and b == 1):
new_one = 0
new_zero = 2 * zero
zero = new_zero
one = new_one
#print(zero, one)
if (s_ ^ 0 == x_):
zero -= 2
print(zero)
``` | output | 1 | 54,413 | 22 | 108,827 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,414 | 22 | 108,828 |
Tags: dp, math
Correct Solution:
```
s, x = map(int, input().split())
print(0 if s < x or (s - x) & (2 * x + 1) else 2 ** bin(x).count('1') - 2 * (s == x))
# Made By Mostafa_Khaled
``` | output | 1 | 54,414 | 22 | 108,829 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Two positive integers a and b have a sum of s and a bitwise XOR of x. How many possible values are there for the ordered pair (a, b)?
Input
The first line of the input contains two integers s and x (2 ≤ s ≤ 1012, 0 ≤ x ≤ 1012), the sum and bitwise xor of the pair of positive integers, respectively.
Output
Print a single integer, the number of solutions to the given conditions. If no solutions exist, print 0.
Examples
Input
9 5
Output
4
Input
3 3
Output
2
Input
5 2
Output
0
Note
In the first sample, we have the following solutions: (2, 7), (3, 6), (6, 3), (7, 2).
In the second sample, the only solutions are (1, 2) and (2, 1). | instruction | 0 | 54,415 | 22 | 108,830 |
Tags: dp, math
Correct Solution:
```
def solve1(s, x):
if x == 0:
if s % 2 == 0:
return 1
else:
return 0
else:
b = (s + x) // 2
a = s - b
if a + b == s and a ^ b == x:
pw = 0
full = (s == x)
for i in range(60):
if x & (1 << i) != 0:
pw += 1
if 2 ** i - 1 == s:
full = True
ans = 2 ** pw
if full:
ans -= 2
return ans
else:
return 0
def solve2(s, x):
cnt = 0
for i in range(1, s):
j = s - i
if i ^ j == x:
cnt += 1
return cnt
# from random import randint
# while True:
# s = randint(0, 10**3)
# x = randint(0, 10**3)
# if solve1(s, x) != solve2(s, x):
# print(s, x, solve1(s, x), solve2(s, x))
# break
s, x = map(int, input().split())
print(solve1(s, x))
``` | output | 1 | 54,415 | 22 | 108,831 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,442 | 22 | 108,884 |
Tags: math, number theory
Correct Solution:
```
def gcd(a, b):
if a==0:
return (b, 0, 1)
g, x1, y1 = gcd(b%a, a)
x = y1 - (b // a) * x1
y = x1
return (g, x, y)
def solve(a, b, x, y, r):
k = (r-x)//a
y = (y-x) % b
gg, X, Y = gcd(a, b)
#print(gg, X, Y, y, a, b)
if y % gg != 0:
return 0
X *= y // gg
dd = b//gg
if X >= 0:
X -= (X//dd) * dd
else:
g = X//dd
if g * dd > X:
g += 1
X -= g * dd
if X < 0:
X += dd
elif X >= dd:
X -= dd
if X > k:
return 0
return (k-X)//dd + 1
a1, b1, a2, b2, L, R = map(int, input().split())
d1 = (L-b1)//a1
if d1 < 0:
d1 = 0
d1 *= a1
d1 += b1
d2 = (L-b2)//a2
if d2 < 0:
d2 = 0
d2 *= a2
d2 += b2
while d1 < L:
d1 += a1
while d2 < L:
d2 += a2
#print(d1, d2, L, R)
if R < max(d1, d2):
print(0)
else:
if d1 > d2 or (d1 == d2 and a1 < a2):
print(solve(a1, a2, d1, d2, R))
else:
print(solve(a2, a1, d2, d1, R))
``` | output | 1 | 54,442 | 22 | 108,885 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,443 | 22 | 108,886 |
Tags: math, number theory
Correct Solution:
```
from collections import defaultdict
import sys, os, math
def gcd(a1, a2):
if a2 == 0:
return a1
else:
return gcd(a2, a1 % a2)
# return (g, x, y) a*x + b*y = gcd(x, y)
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, x, y = egcd(b % a, a)
return (g, y - (b // a) * x, x)
if __name__ == "__main__":
#n, m = list(map(int, input().split()))
a1, b1, a2, b2, L, R = map(int, input().split())
a2 *= -1
LCM = a1 * a2 // gcd(a1, a2)
if abs(b1 - b2) % gcd(a1, a2) != 0:
print(0)
sys.exit(0)
L = max([b1, b2, L])
g, x, y = egcd(a1, a2)
X = a1 * x * (b2 - b1) // g + b1
X += LCM * math.ceil((L - X) / LCM)
if L <= X <= R:
print(max(0, (R - X) // LCM + 1))
else:
print(0)
``` | output | 1 | 54,443 | 22 | 108,887 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,444 | 22 | 108,888 |
Tags: math, number theory
Correct Solution:
```
def extgcd(a, b):
x, y = 0, 0
d = a;
if b != 0:
d, y, x = extgcd(b, a%b)
y -= (a//b) * x
else:
x, y = 1, 0
return (d, x, y)
def main():
a1, b1, a2, b2, L, R = map(int, input().split())
g, k, l = extgcd(a1, a2);
b = b2-b1;
if (b%g != 0):
print (0)
return
k *= b//g
l *= -b//g
low = -2**100
high = 2**100
while high-low > 1:
med = (low+high)//2
tk = k+med*a2//g
tl = l+med*a1//g
if (tk >= 0 and tl >= 0):
high = med
else:
low = med
k = k+high*a2//g
x = a1*k+b1
low = -1
high = 2**100
lcm = a1*a2//g
while high - low > 1:
med = (low+high)//2
tx = x+med*lcm
if tx >= L:
high = med
else:
low = med
x = x+high*lcm
low = 0
high = 2**100
while high-low > 1:
med = (low+high)//2
tx = x+med*lcm
if (tx <= R):
low = med
else:
high = med
if low == 0 and x > R:
print (0)
return
print (low+1)
return
if __name__=="__main__":
main()
``` | output | 1 | 54,444 | 22 | 108,889 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,445 | 22 | 108,890 |
Tags: math, number theory
Correct Solution:
```
import sys
# Uz ma to pretekanie nebavi!!!
def gcd(a, b):
if b == 0:
return [a, 1, 0]
c = a%b
[g, x1, y1] = gcd(b, c)
x = y1
y = x1 - y1 * (a//b)
return [g, x, y]
a1, b1, a2, b2, l, r = [int(i) for i in input().split(" ")]
if max(b1, b2) > r:
print(0)
sys.exit(0)
l = max(l, b1, b2)
[g, xg, yg] = gcd(a1, a2)
if (b2 - b1) % g == 0:
xg *= (b2 - b1) // g
else:
print(0)
sys.exit(0)
lcm = (a1 * a2) // g
val = xg * a1 + b1
if val >= l:
val -= (((val - l) // lcm) + 1) * lcm
print(((r - val) // lcm) - ((l - val - 1) // lcm))
``` | output | 1 | 54,445 | 22 | 108,891 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,446 | 22 | 108,892 |
Tags: math, number theory
Correct Solution:
```
def nod(a, b):
if b == 0:
return a, 1, 0
else:
answer, x, y = nod(b, a % b)
x1 = y
y1 = x - (a // b) * y
return answer, x1, y1
a1, b1, a2, b2, l, r = list(map(int, input().split()))
coeff = b1
b1, b2, l, r = b1 - coeff, b2 - coeff, max(l - coeff, 0), r - coeff
l = max(b2, l)
od, x1, y1 = nod(a1, -a2)
if b2 % od != 0 or l > r:
print(0)
else:
x1, y1 = x1 * (b2 // od), y1 * (b2 // od)
result = x1 * a1
raznitsa = a1 * a2 // nod(a1, a2)[0]
otvet = 0
if result < l:
vsp = (l - result) // raznitsa
if (l - result) % raznitsa != 0:
vsp += 1
result += vsp * raznitsa
if result > r:
vsp = (result - r) // raznitsa
if (result - r) % raznitsa != 0:
vsp += 1
result -= vsp * raznitsa
if result <= r and result >= l:
otvet += 1
otvet += abs(result - r) // raznitsa
otvet += abs(result - l) // raznitsa
print(otvet)
# 3 * (- 54) + 81 =
``` | output | 1 | 54,446 | 22 | 108,893 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,447 | 22 | 108,894 |
Tags: math, number theory
Correct Solution:
```
#from IPython import embed
def xgcd(b, n):
x0, x1, y0, y1 = 1, 0, 0, 1
while n != 0:
q, b, n = b // n, n, b % n
x0, x1 = x1, x0 - q * x1
y0, y1 = y1, y0 - q * y1
return b, x0, y0
def ffloor(a, b):
if(b < 0): return ffloor(-a,-b);
return a//b
def cceil( a, b):
if(b < 0): return cceil(-a,-b);
if a % b == 0:
return a//b
return a//b+1;
def main():
s = input()
a1, b1, a2, b2, L, R = [int(i) for i in s.split()]
if b2 < b1:
a1, a2 , b1, b2 = a2, a1 , b2, b1
d,x,y = xgcd(a1,-a2)#extended_gcd(a1,-a2)
if(d < 0):
d *= -1
x *= -1
y *= -1
if (b2 - b1) % d != 0:
print(0)
return
#print(d,x,y)
fact = (b2-b1)//d
x *= fact
y *= fact
c1 = a2//d;
c2 = a1//d;
tope1 = ffloor(R-b1-a1*x, a1*c1);
bajo1 = cceil(L-b1-a1*x,c1*a1);
bajo2 = cceil(L-b2-a2*y,c2*a2);
tope2 = ffloor(R-b2-a2*y, a2*c2);
bajo3 = max(cceil(-x,c1),cceil(-y,c2));
bajo = max(bajo1,bajo2,bajo3);
tope = min(tope1,tope2);
print(max(0,tope+1-bajo))
#embed()
main()
``` | output | 1 | 54,447 | 22 | 108,895 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,448 | 22 | 108,896 |
Tags: math, number theory
Correct Solution:
```
import sys
def gcd(x,y):
if y==0: return (x,1,0)
k = x//y
g,a,b = gcd(y,x%y)
return (g,b,a-b*k)
read = lambda: (int(x) for x in sys.stdin.readline().split())
a1,b1,a2,b2,l,r = read()
l = max(l,b1,b2)
r = max(l,r+1)
g,x,y = gcd(a1,a2)
if (b1-b2)%g: print(0), exit(0)
a = a1*a2 // g
b = b1 + (b2-b1)//g * x * a1
b %= a
lk = l // a + (l%a > b)
rk = r // a + (r%a > b)
print(rk-lk)
``` | output | 1 | 54,448 | 22 | 108,897 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2 | instruction | 0 | 54,449 | 22 | 108,898 |
Tags: math, number theory
Correct Solution:
```
import sys, collections
def gcd(a, b):
if b == 0: return a
return gcd(b, a % b)
def lcm(a, b):
return a // gcd(a, b) * b
def extgcd(a, b):
if b == 0: return 1, 0
x, y = extgcd(b, a % b)
return y, x - a // b * y
def prime_factor(n):
res = collections.defaultdict(int)
i = 2
while i * i <= n:
cnt = 0
while n % i == 0:
n //= i
cnt += 1
if cnt > 0: res[i] = cnt
i += 1
if n != 1: res[n] = 1
return res
def modinv(a, mod):
if a == 0: return -1
if gcd(a, mod) != 1: return -1
return extgcd(a, mod)[0] % mod
def normalize(a1, a2):
p1 = prime_factor(a1)
p2 = prime_factor(a2)
keys = list(set(p1.keys()) | set(p2.keys()))
r1 = 1
r2 = 1
for k in keys:
if p1[k] >= p2[k]:
r1 *= k ** p1[k]
else:
r2 *= k ** p2[k]
return r1, r2
def solve(a1, b1, a2, b2):
g = gcd(a1, a2)
if (b1 - b2) % g != 0: return -1
a1, a2 = normalize(a1, a2)
u = b1 % a1
inv = modinv(a1, a2)
v = (b2 - u) * inv % a2
return u + v * a1
def f(x0, T, v):
ok = 10 ** 36
ng = -1
while ok - ng > 1:
mid = (ok + ng) // 2
if x0 + T * mid >= v:
ok = mid
else:
ng = mid
return ok
a1, b1, a2, b2, L, R = map(int, input().split())
T = lcm(a1, a2)
x0 = solve(a1, b1, a2, b2)
if x0 == -1:
print(0)
sys.exit()
x0 -= T * 10 ** 36
ok = 10 ** 60
ng = -1
while ok - ng > 1:
mid = (ok + ng) // 2
val = x0 + T * mid
k = (val - b1) // a1
l = (val - b2) // a2
if k >= 0 and l >= 0:
ok = mid
else:
ng = mid
x0 += ok * T
# L <= x0 + kT < R + 1
ans = f(x0, T, R + 1) - f(x0, T, L)
print(ans)
``` | output | 1 | 54,449 | 22 | 108,899 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
a1, b1, a2, b2, L, R = map(int, input().split())
def xgcd(a,b):
prevx, x = 1, 0
prevy, y = 0, 1
while b:
q = a // b
x, prevx = prevx - q * x, x
y, prevy = prevy - q * y, y
a, b = b, a % b
return a, prevx, prevy
g, x, y = xgcd(a1, -a2)
if (b2 - b1) // g < 0:
g, x, y = -g, -x, -y
if abs(b2 - b1) % abs(g) > 0:
print(0)
else:
a2g, a1g = a2 // abs(g), a1 // abs(g)
x *= (b2 - b1) // g
y *= (b2 - b1) // g
if x < 0:
y += ((abs(x) + a2g - 1) // a2g) * a1g
x += ((abs(x) + a2g - 1) // a2g) * a2g
if y < 0:
x += ((abs(y) + a1g - 1) // a1g) * a2g
y += ((abs(y) + a1g - 1) // a1g) * a1g
if x >= 0 and y >= 0:
k = min(x // a2g, y // a1g)
x -= k * a2g
y -= k * a1g
res = a1 * x + b1
lcm = a1 * a2 // abs(g)
L, R = max(0, L - res), R - res
if R < 0:
print(0)
else:
print(R // lcm - L // lcm + (L % lcm == 0))
``` | instruction | 0 | 54,450 | 22 | 108,900 |
Yes | output | 1 | 54,450 | 22 | 108,901 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
import math
# g, x, y
def gcd(a, b) :
if a == 0 :
return [b, 0, 1]
l = gcd(b % a, a)
g, x1, y1 = [int(i) for i in l]
x = y1 - (b // a) * x1
y = x1
return [g, x, y]
def my_ceil(u, v) :
if v < 0 :
u *= -1
v *= -1
return math.ceil(u / v)
def my_floor(u, v) :
if v < 0 :
u *= -1
v *= -1
return math.floor(u / v)
a1, b1, a2, b2, L, R = [int(i) for i in input().split()]
A = a1
B = -a2
C = b2 - b1
g, x0, y0 = [int(i) for i in gcd(abs(A), abs(B))]
if A < 0 : x0 *= -1
if B < 0 : y0 *= -1
if C % g != 0 :
print(0)
exit()
x0 *= C // g
y0 *= C // g
le = max([
float(R - b1 - a1 * x0) / float(a1 * B // g),
float(y0 * a2 + b2 - R) / float(a2 * A // g)
])
ri = min([
float(L - b1 - a1 * x0) / float(a1 * B // g),
float(y0 * a2 + b2 - L) / float(a2 * A // g),
float(-x0) / float(B // g),
float(y0) / float(A // g)
])
le = int(math.ceil(le))
ri = int(math.floor(ri))
if ri - le + 1 <= 10000 :
result = 0
for k in range(le - 100, ri + 101) :
X = x0 + B * k // g
Y = y0 - A * k // g
if X >= 0 and Y >= 0 and a1 * X + b1 >= L and a1 * X + b1 <= R :
result += 1
print(result)
else :
print(max(int(0), ri - le + 1))
``` | instruction | 0 | 54,451 | 22 | 108,902 |
Yes | output | 1 | 54,451 | 22 | 108,903 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
def exgcd(i, j):
if j == 0:
return 1, 0, i
u, v, d = exgcd(j, i % j)
return v, u - v * (i // j), d
ma, ra, mb, rb, L, R = map(int, input().split(' '))
L = max(L, ra, rb)
if L > R:
print(0)
exit(0)
if ra > rb:
ma, ra, mb, rb = mb, rb, ma, ra
_, _, md = exgcd(ma, mb)
if md != 1:
if (rb - ra) % md != 0:
print(0)
exit(0)
m = ma * mb // md
rev, _, _ = exgcd(ma // md, mb // md)
rev = (rev % (mb // md) + mb // md) % (mb // md)
r = ma * (rb - ra) // md * rev + ra
r = (r % m + m) % m
else:
m = ma * mb
bv, av, _ = exgcd(ma, mb)
r = ra * mb * av + rb * ma * bv
r = (r % m + m) % m
def calc(i):
return (i - r) // m
print(calc(R) - calc(L - 1))
``` | instruction | 0 | 54,452 | 22 | 108,904 |
Yes | output | 1 | 54,452 | 22 | 108,905 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
#from IPython import embed
def mod(a, b):
if b < 0:
return mod(a,-b)
if a >= 0:
return a % b
return - ((-a)%b)
def extended_gcd(a, b):
tmp1 = a
tmp2 = b
xx = 0
y = 0
yy = 1
x = 1
while b != 0:
q = a//b
t = b
b = mod(a,b)
a = t
tt = xx
xx = x-q*xx
x = t
t = yy
yy = y-q*yy
y = t;
assert(a == tmp1*x+tmp2*y)
return (a,x,y)
def xgcd(b, n):
x0, x1, y0, y1 = 1, 0, 0, 1
while n != 0:
q, b, n = b // n, n, b % n
x0, x1 = x1, x0 - q * x1
y0, y1 = y1, y0 - q * y1
return b, x0, y0
def ffloor(a, b):
if(b < 0): return ffloor(-a,-b);
return a//b
def cceil( a, b):
if(b < 0): return cceil(-a,-b);
if a % b == 0:
return a//b
return a//b+1;
def main():
s = input()
a1, b1, a2, b2, L, R = [int(i) for i in s.split()]
if b2 < b1:
a1, a2 , b1, b2 = a2, a1 , b2, b1
d,x,y = xgcd(a1,-a2)#extended_gcd(a1,-a2)
if(d < 0):
d *= -1
x *= -1
y *= -1
if (b2 - b1) % d != 0:
print(0)
return
#print(d,x,y)
fact = (b2-b1)//d
x *= fact
y *= fact
c1 = a2//d;
c2 = a1//d;
tope1 = ffloor(R-b1-a1*x, a1*c1);
bajo1 = cceil(L-b1-a1*x,c1*a1);
bajo2 = cceil(L-b2-a2*y,c2*a2);
tope2 = ffloor(R-b2-a2*y, a2*c2);
bajo3 = max(cceil(-x,c1),cceil(-y,c2));
#print((R-b1-a1*x) /( a1*c1) ,(R-b2-a2*y)/ (a2*c2))
#print((L-b1-a1*x)/(c1*a1) ,(L-b2-a2*y)/(c2*a2))
#print(-x/c1,-y/c2)
#print(bajo1,tope1)
#print(bajo2,tope2)
#print(bajo3)
bajo = max(bajo1,bajo2,bajo3);
tope = min(tope1,tope2);
print(max(0,tope+1-bajo))
#embed()
main()
``` | instruction | 0 | 54,453 | 22 | 108,906 |
Yes | output | 1 | 54,453 | 22 | 108,907 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
from math import gcd
def exd_gcd(a, b):
# always return as POSITIVE presentation
if a % b == 0:
return 0, (1 if b > 0 else -1)
x, y = exd_gcd(b, a % b)
return y, x - a // b * y
def interval_intersect(a, b, c, d):
if b <= a or d <= c:
return 0
if c < a:
a, b, c, d = c, d, a, b
if c < b:
return min(b, d) - c
else:
return 0
def ceil(a, b):
return (a + b - 1) // b
a1, b1, a2, b2, L, R = map(int, input().split())
g = gcd(a1, a2)
if (b1 - b2) % g != 0:
print(0)
exit(0)
k, l = exd_gcd(a1, a2)
l = -l
k *= (b2 - b1) // g
l *= (b2 - b1) // g
d1 = a2 // g
d2 = a1 // g
assert(k * a1 + b1 == l * a2 + b2)
arb = 3238
assert((k + arb * d1) * a1 + b1 == (l + arb * d2) * a2 + b2)
L1, R1 = ceil(max(0, ceil(L - b1, a1)) - k, d1), ((R - b1) // a1 - k) // d1
L2, R2 = ceil(max(0, ceil(L - b2, a2)) - k, d2), ((R - b2) // a2 - k) // d2
print(interval_intersect(L1, R1 + 1, L2, R2 + 1))
``` | instruction | 0 | 54,454 | 22 | 108,908 |
No | output | 1 | 54,454 | 22 | 108,909 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
def gcd(a, b):
if a==0:
return (b, 0, 1)
g, x1, y1 = gcd(b%a, a)
x = y1 - (b // a) * x1
y = x1
return (g, x, y)
def solve(a, b, x, y, r):
k = (r-x)//a
y = (-y) % b
y = (y-x) % b
gg, X, Y = gcd(a, b)
if y % gg != 0:
return 0
X *= y // gg
dd = b//gg
X -= (X//dd) * dd
if X < 0:
X += dd
elif X >= dd:
X -= dd
if X > k:
return 0
return (k-X)//dd + 1
a1, b1, a2, b2, L, R = map(int, input().split())
d1 = (L-b1)//a1
if d1 < 0:
d1 = 0
d1 *= a1
d1 -= b1
d2 = (L-b2)//a2
if d2 < 0:
d2 = 0
d2 *= a2
d2 -= b2
while d1 < L:
d1 += a1
while d2 < L:
d2 += a2
#print(d1, d2, L, R)
if R < max(d1, d2):
print(0)
else:
if d1 > d2 or (d1 == d2 and a1 < a2):
print(solve(a1, a2, d1, d2, R))
else:
print(solve(a2, a1, d2, d1, R))
``` | instruction | 0 | 54,455 | 22 | 108,910 |
No | output | 1 | 54,455 | 22 | 108,911 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
from collections import defaultdict
import sys, os, math
def gcd(a1, a2):
if a2 == 0:
return a1
else:
return gcd(a2, a1 % a2)
# return (g, x, y) a*x + b*y = gcd(x, y)
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, x, y = egcd(b % a, a)
return (g, y - (b // a) * x, x)
if __name__ == "__main__":
#n, m = list(map(int, input().split()))
a1, b1, a2, b2, L, R = map(int, input().split())
a2 *= -1
LCM = a1 * a2 // gcd(a1, a2)
if abs(b1 - b2) % gcd(a1, a2) != 0:
print(0)
sys.exit(0)
L = max([b1, b2, L])
g, x, y = egcd(a1, a2)
X = a1 * x * (b2 - b1) // g + b1
X += LCM * math.ceil((L - X) / LCM)
print(max(0, (R - X) // LCM + 1))
``` | instruction | 0 | 54,456 | 22 | 108,912 |
No | output | 1 | 54,456 | 22 | 108,913 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.
Input
The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109, - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).
Output
Print the desired number of integers x.
Examples
Input
2 0 3 3 5 21
Output
3
Input
2 4 3 0 6 17
Output
2
Submitted Solution:
```
def xgcd (b,n) :
x0,x1,y0,y1 = 1,0,0,1
while n != 0 :
q,b,n = b//n , n , b % n
x0,x1 = x1, x0-q*x1
y0,y1 = y1,y0-q*y1
return b,x0,y0
a,aa,b,bb,l,r = [int (x) for x in input ().split ()]
g,x,y=xgcd (a,b)
c = bb-aa
if c%g != 0 :
print (0)
exit (0)
i = a*(x*c)+aa
step = a*b//g
# shift i to la,lb
if i >= max(aa,bb) :
i -= (i-max(aa,bb))//step * step
else :
i += ((max(aa,bb)-i)//step + 1) * step
f = (l-i)//step
if (l-i) % step != 0 : f+=1
s = (r-i)//step
#print(i,step,f,s)
print (max(0,s-f+1))
``` | instruction | 0 | 54,457 | 22 | 108,914 |
No | output | 1 | 54,457 | 22 | 108,915 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers a and b. Moreover, you are given a sequence s_0, s_1, ..., s_{n}. All values in s are integers 1 or -1. It's known that sequence is k-periodic and k divides n+1. In other words, for each k ≤ i ≤ n it's satisfied that s_{i} = s_{i - k}.
Find out the non-negative remainder of division of ∑ _{i=0}^{n} s_{i} a^{n - i} b^{i} by 10^{9} + 9.
Note that the modulo is unusual!
Input
The first line contains four integers n, a, b and k (1 ≤ n ≤ 10^{9}, 1 ≤ a, b ≤ 10^{9}, 1 ≤ k ≤ 10^{5}).
The second line contains a sequence of length k consisting of characters '+' and '-'.
If the i-th character (0-indexed) is '+', then s_{i} = 1, otherwise s_{i} = -1.
Note that only the first k members of the sequence are given, the rest can be obtained using the periodicity property.
Output
Output a single integer — value of given expression modulo 10^{9} + 9.
Examples
Input
2 2 3 3
+-+
Output
7
Input
4 1 5 1
-
Output
999999228
Note
In the first example:
(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = 2^{2} 3^{0} - 2^{1} 3^{1} + 2^{0} 3^{2} = 7
In the second example:
(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = -1^{4} 5^{0} - 1^{3} 5^{1} - 1^{2} 5^{2} - 1^{1} 5^{3} - 1^{0} 5^{4} = -781 ≡ 999999228 \pmod{10^{9} + 9}. | instruction | 0 | 54,518 | 22 | 109,036 |
Tags: math, number theory
Correct Solution:
```
MOD = 1000000009
def inv(n): # Calculate inverse of n modulo MOD
return pow(n, MOD - 2, MOD)
n, a, b, k = map(int, input().split()) # first line of input
q = (n + 1) // k
string = input() # second line of input
s = [] # list for storing +1 and -1 as and when + or - occur in the string
for char in string:
if char == "+": s.append(1)
else: s.append(-1)
res = 0 # final answer
for i in range(k): # Calculating the sum till first period, i.e. from s_0 to s_(k-1)
res += (s[i] * pow(a, n - i, MOD) * pow(b, i, MOD))
res %= MOD
# For checking if the sum R := 1 + T + T^2 + ... + T^(q-1) = (T^q - 1) / (T - 1) for T = (b/a)^k is divisible by MOD
# becuase then the inverse of d1 modulo MOD doesn't exist (in this notation, T would be 1 modulo MOD, so the sum becomes 0/0 when modulo of numerator and denominator are taken)
# and so the code in the following "if" part won't be valid in this case
n1 = pow(b, k, MOD)
n2 = pow(a, k, MOD)
n2 = inv(n2)
T = n1 * n2 % MOD
if T != 1: # when inverse of d1 modulo MOD exists (and the sum (T^q - 1)/(T - 1) is not of the form 0/0)
num = (pow(b, n + 1, MOD) - pow(a, n + 1, MOD)) % MOD # numerator
d1 = (pow(b, k, MOD) - pow(a, k, MOD)) % MOD
d1 = inv(d1) # one part of the denominator
d2 = pow(a, n + 1 - k, MOD)
d2 = inv(d2) # other part of the denominator
R = (num * d1 * d2) % MOD # the sum R defined above in comments, modulo MOD
res = (res * R) % MOD # final answer = res * R
print(res)
else: # when T = 1, the sum 1 + T + T^2 + ... + T^(q-1) is simply q * T = q and so R = q
res *= q # final answer = res * R
res %= MOD
print(res)
``` | output | 1 | 54,518 | 22 | 109,037 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers a and b. Moreover, you are given a sequence s_0, s_1, ..., s_{n}. All values in s are integers 1 or -1. It's known that sequence is k-periodic and k divides n+1. In other words, for each k ≤ i ≤ n it's satisfied that s_{i} = s_{i - k}.
Find out the non-negative remainder of division of ∑ _{i=0}^{n} s_{i} a^{n - i} b^{i} by 10^{9} + 9.
Note that the modulo is unusual!
Input
The first line contains four integers n, a, b and k (1 ≤ n ≤ 10^{9}, 1 ≤ a, b ≤ 10^{9}, 1 ≤ k ≤ 10^{5}).
The second line contains a sequence of length k consisting of characters '+' and '-'.
If the i-th character (0-indexed) is '+', then s_{i} = 1, otherwise s_{i} = -1.
Note that only the first k members of the sequence are given, the rest can be obtained using the periodicity property.
Output
Output a single integer — value of given expression modulo 10^{9} + 9.
Examples
Input
2 2 3 3
+-+
Output
7
Input
4 1 5 1
-
Output
999999228
Note
In the first example:
(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = 2^{2} 3^{0} - 2^{1} 3^{1} + 2^{0} 3^{2} = 7
In the second example:
(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = -1^{4} 5^{0} - 1^{3} 5^{1} - 1^{2} 5^{2} - 1^{1} 5^{3} - 1^{0} 5^{4} = -781 ≡ 999999228 \pmod{10^{9} + 9}. | instruction | 0 | 54,519 | 22 | 109,038 |
Tags: math, number theory
Correct Solution:
```
n, a, b, k = [int(i) for i in input().split()]
st = input()
l = (n + 1) // k
s = 0
mod = 1000000009
def f_pow(a, k):
if k == 0:
return 1
if k % 2 == 1:
return f_pow(a, k - 1) * a % mod
else:
return f_pow(a * a % mod, k // 2) % mod
def rev(b):
return f_pow(b, mod - 2)
q = f_pow(b, k) * rev(f_pow(a, k))
qn = f_pow(q, l)
rq = rev(q - 1)
g1 = f_pow(a, n)
ra = rev(a)
for i in range(len(st)):
sgn = 1 - 2 * (st[i] == '-')
res = g1 * (qn - 1) * rq
if (q % mod) != 1:
s = (s + sgn * res) % mod
else:
s = (s + sgn * g1 * l) % mod
g1 = g1 * ra * b % mod
print(s)
``` | output | 1 | 54,519 | 22 | 109,039 |
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