Problem:
Score : 500 points

Problem Statement

For integers b (b \geq 2) and n (n \geq 1), let the function f(b,n) be defined as follows:

f(b,n) = n, when n < b

f(b,n) = f(b,\,{\rm floor}(n / b)) + (n \ {\rm mod} \ b), when n \geq b

Here, {\rm floor}(n / b) denotes the largest integer not exceeding n / b,
and n \ {\rm mod} \ b denotes the remainder of n divided by b.

Less formally, f(b,n) is equal to the sum of the digits of n written in base b.
For example, the following hold:

f(10,\,87654)=8+7+6+5+4=30

f(100,\,87654)=8+76+54=138

You are given integers n and s.
Determine if there exists an integer b (b \geq 2) such that f(b,n)=s.
If the answer is positive, also find the smallest such b.

Constraints

1 \leq n \leq 10^{11}

1 \leq s \leq 10^{11}

n,\,s are integers.

Input

The input is given from Standard Input in the following format:

n
s

Output

If there exists an integer b (b \geq 2) such that f(b,n)=s, print the smallest such b.
If such b does not exist, print -1 instead.

Sample Input 1

87654
30

Sample Output 1

10

Sample Input 2

87654
138

Sample Output 2

100

Sample Input 3

87654
45678

Sample Output 3

-1

Sample Input 4

31415926535
1

Sample Output 4

31415926535

Sample Input 5

1
31415926535

Sample Output 5

-1

Solution in Python (3.4.3) (judge verdict: Time Limit Exceeded)

import sys
sys.stdin.readline

def digitsum(b, n):
    ans = 0
    while n:
        n, m = divmod(n, b)
        ans += m
    return ans

def main():
    n = int(input())
    s = int(input())
    if s == n:
        print(n+1)
        return
    if s > n:
        print(-1)
        return
    for b in range(2, int(n**.5)+1):
        if digitsum(b, n) == s:
            print(b)
            return
    ans = float("inf")
    for d in range(1, int(n**.5)+1):
        b = (n - s) // d + 1
        if digitsum(b, n) == s:
            ans = min(ans, b)
    if ans == float("inf"):
        ans = -1
    print(ans)

if __name__ == "__main__":
    main()