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 (2.11.7) (judge verdict: Wrong Answer)

import scala.io.StdIn

object Main {

  def main(args: Array[String]): Unit = {

    val n = StdIn.readLong()
    val s = StdIn.readLong()

    val n2 = math.sqrt(n).toLong

    var found = -1L

    if (n == s) {
      found = n + 1L
      println(found)
    }

    for (i <- 2L to n2 if found < 0L) {
      if (f(i, n) == s) {
        found = i
        println(found)
      }
    }

    if (found < 0L) {
      for (i <- (n2 - 1L).to(1L, -1L) if found < 0L) {
        val b1 = n / (i + 1) + 1
        val a1 = n % b1 + i
        val max = n / i

        findInSequence(s, a1, -i) match {
          case Some(j) if j > 0 =>
            val b = b1 + (j - 1)
            if (b <= max) {
              found = b
            }
          case _ =>
        }
      }

      println(found)
    }
  }

  def findInSequence(x: Long, a1: Long, d: Long): Option[Long] = {
    val y = x - a1 + d
    if (y % d == 0L) Some(y / d) else None
  }

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