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.Arrays;
import java.util.Scanner;

public class Main {
  Scanner sc = new Scanner(System.in);

  public static void main(String[] args) {
    new Main().run();
  }

  long f(long b, long n) {
    if (n < b) {
      return n;
    } else {
      return f(b, n / b) + n % b;
    }
  }

  void run() {
    long n = sc.nextLong();
    long s = sc.nextLong();
    if (n == s) {
      System.out.println(n + 1);
      return;
    }
    long ans = Long.MAX_VALUE;
    for (long b = 2; b * b <= n; ++b) {
      if (f(b, n) == s) {
        ans = Math.min(ans, b);
      }
    }
    for (long p = 1; p * p < n; ++p) {
      long b = (n - s) / p + 1;
      if (b < 2) {
        continue;
      }
      if (f(b, n) == s) {
        ans = Math.min(ans, b);
      }
    }
    if (ans != Long.MAX_VALUE) {
      System.out.println(ans);
    } else {
      System.out.println(-1);
    }
  }

  int ni() {
    return Integer.parseInt(sc.next());
  }

  void debug(Object... os) {
    System.err.println(Arrays.deepToString(os));
  }
}