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: Accepted)

object Main extends App {

  import scala.io.StdIn._

  val n = readLong()
  val s = readLong()

  val rt = Math.sqrt(n).toLong

  def digitSum(b : Long, n : Long) : Long = if (n < b) n else digitSum(b , n / b) + n % b

  def checkOne = if (n == s) Some(n + 1) else None

  def searchFirstHalf(now : Long):Option[Long] = {
    if (now >= rt)  None
    else if (now > 1 && digitSum(now, n) == s) Some(now)
    else searchFirstHalf(now + 1)
  }

  def searchSecondHalf(now : Long) : Option[Long] = {
    if (now <= 0) None
    else {
      val b = (n + now - s) / now
      if (b > 1 && digitSum(b, n) == s) {
        Some(b)
      }
      else searchSecondHalf(now - 1)
    }
  }

  val calc = if (n < s) None else checkOne orElse searchFirstHalf(2) orElse searchSecondHalf(rt)

  println(calc getOrElse -1)
}