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 Kotlin (1.0.0) (judge verdict: Accepted)

import java.util.*
import kotlin.*

import java.io.InputStream
import java.lang.NumberFormatException
import java.lang.StringBuilder


fun main(args: Array<String>) {
    Program().solve()
}

class Program {

    fun f(b: Long, n: Long): Long = if (n < b) n else f(b, n / b) + n % b

    fun solve() {
        val sc = FastScanner()
        val n = sc.nextLong()
        val s = sc.nextLong()

        // b <= √n nは三桁以上 実際に計算
        var a = 2L

        while (a * a <= n) {
            if (f(a, n) == s) {
                println(a)
                return
            }
            a++
        }

        // √n < b <= n nは2桁
        // s = floor(n/b) + n%b
        // floor(n/b) * b = n-(s-floor(n/b))
        // b = n-(s-floor(n/b)) /(floor(n/b))
        for (nPerB in a - 1 downTo 1) {
            val t = n - (s - nPerB)
            if (t % nPerB == 0L) {
                val b = t / nPerB
                if (b in a..n && f(b, n) == s) {
                    println(b)
                    return
                }
            }
        }

        // n < b nは1桁
        // b = n+1 (s == n)

        if (s == n) {
            println(n + 1)
            return
        }

        println(-1)
    }
}


class FastScanner {
    companion object {
        val input: InputStream = System.`in`
        val buffer = ByteArray(1024) { 0 }
        fun isPrintableChar(c: Int): Boolean = c in 33..126
    }

    var ptr = 0
    var buflen = 0
    private fun hasNextByte(): Boolean {
        if (ptr < buflen) {
            return true
        } else {
            ptr = 0
            buflen = input.read(buffer)
            if (buflen <= 0) {
                return false
            }
        }
        return true
    }

    private fun readByte(): Int = if (hasNextByte()) buffer[ptr++].toInt() else -1

    private fun skipUnprintable() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr].toInt())) ptr++
    }

    fun hasNext(): Boolean {
        skipUnprintable()
        return hasNextByte()
    }

    fun next(): String {
        if (!hasNext()) throw NoSuchElementException()
        val sb = StringBuilder()
        var b = readByte()
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b)
            b = readByte()
        }
        return sb.toString()
    }

    fun nextInt(): Int {
        if (!hasNext()) throw NoSuchElementException()
        var n = 0
        var b = readByte()
        // '-' = 45
        val minus = b == 45
        if (minus) {
            b = readByte()
        }

        // '0' = 48 '9' = 57
        if (b !in 48..57) {
            throw NumberFormatException()
        }

        while (true) {
            if (b in 48..57) {
                n *= 10
                n += b - 48
            } else if (b == -1 || !isPrintableChar(b)) {
                return if (minus) -n else n
            } else {
                throw NumberFormatException()
            }
            b = readByte()
        }
    }

    fun nextLong(): Long {
        if (!hasNext()) throw NoSuchElementException()
        var n = 0L
        var b = readByte()
        // '-' = 45
        val minus = b == 45
        if (minus) {
            b = readByte()
        }

        // '0' = 48 '9' = 57
        if (b !in 48..57) {
            throw NumberFormatException()
        }

        while (true) {
            if (b in 48..57) {
                n *= 10
                n += b - 48
            } else if (b == -1 || !isPrintableChar(b)) {
                return if (minus) -n else n
            } else {
                throw NumberFormatException()
            }
            b = readByte()
        }
    }

    fun nextDouble(): Double = next().toDouble()
}