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 Scala (2.13.1) (judge verdict: Accepted)

object Main extends App {
  val sc = new java.util.Scanner(System.in)
  val n, s = sc.nextLong()

  def f(b: Long, n: Long, res: Long = 0): Long = {
    if (n == 0) res else f(b, n/b, res + (n % b))
  }

  if (n < s) {
    println(-1)
  } else if (n == s) {
    println(n+1)
  } else {
    var b = 2L
    var continue = true
    while (continue && b * b <= n) {
      if (f(b, n) == s) {
        println(b)
        continue = false
      }
      b += 1
    }

    if (continue) {
      var p = 1L
      var ans = -1L
      while (p * p <= n) {
        if ((n-s) % p == 0) {
          val b = (n-s)/p + 1
          if (f(b, n) == s) {
            ans = b
          }
        }
        p += 1
      }

      if (ans != -1) {
        println(ans)
      } else {
        println(-1)
      }
    }
  }
}