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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.Scanner;

/**
 * 解説通り実装したソースコード
 */
public class Main {

	public static void main(String[] args) {
		try (Scanner scanner = new Scanner(System.in)) {
			long n = scanner.nextLong(), s = scanner.nextLong();
			if (n < s) {
				// https://atcoder.jp/contests/abc044/submissions/4095161 参考に追加
				System.out.println(-1);
			} else if (n == s) {
				System.out.println(n + 1);
			} else {
				int sqrtN = (int) Math.sqrt(n);
				for (int b = 2; b <= sqrtN; b++) {
					if (s == f(b, n)) {
						System.out.println(b);
						return;
					}
				}
				for (int p = sqrtN; p >= 1; p--) {
					long b = (n - s) / p + 1;
					if ((b > sqrtN) && (b > p) && (s == f(b, n))) {
						System.out.println(b);
						return;
					}
				}
				System.out.println(-1);
			}
		}
	}

	/**
	 * @param b
	 * @param n
	 * @return f(b,n)
	 */
	private static long f(long b, long n) {
		if (n < b) {
			return n;
		} else {
			return f(b, n / b) + n % b;
		}
	}
}