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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.*;

public class Main {
  public static void main(String[] args) {
    Scanner sc = new Scanner(System.in);
    long n = sc.nextLong();
    long s = sc.nextLong();
    long ans = (long)Math.pow(10, 12);
    if(n == s) {
      ans = n + 1;
    }  else {
    for(long b = 2; (b * b) <= n; b++) {
      long t = n;
      long c = 0;
      while(t > 0) {
        c += (t % b);
        t /= b;
      }
      if(c == s) {
        ans = Math.min(ans, b);
        break;
      }
    }
    for(long p = 1; (p * p) <= n; p++) {
      long q = s - p;
      if((q >= 0) && (n > q) && (((n - q) % p) == 0)) {
        long b = (n - q) / p;
        if((b > p) && (b > q)) {
          long t = n;
          long c = 0;
          while(t > 0) {
            c += (t % b);
            t /= b;
          }
          if(c == s) ans = Math.min(ans, b);
        }
      }
    }
    if(s == 1) ans = Math.min(ans, n);
    if(ans == (long)Math.pow(10, 12)) ans = -1;
    }
    System.out.println(ans);
  }
}