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)

fun main(arg: Array<String>) {
    val N = nextLong()
    val S = nextLong()
    if (N == S) return print(N + 1)
    val sqrtN = Math.sqrt(N.toDouble()).toLong()
    val list = mutableListOf<Long>()
    fun f(b: Long, N: Long): Long {
        var n = N
        var cnt = 0L
        while (n > 0) {
            cnt += n % b
            n /= b
        }
        return cnt
    }
    for (b in 2 .. sqrtN) if (f(b, N) == S) return print(b)
    val A = N - S
    if (A < 0) return print(-1)
    for (n in 1 .. sqrtN) {
        if (A % n == 0L) {
            list.add(n + 1)
            if (n != A / n) list.add(A / n + 1)
        }
    }
    list.sort()
    for (b in list) {
        if (b > sqrtN && f(b, N) == S)
            return print(b)
    }
    return print(-1)
}

fun next() = readLine()!!
fun nextInt() = next().toInt()
fun nextLong() = next().toLong()
fun listOfString() = next().split(" ")
fun listOfInt() = listOfString().map(String::toInt)
fun listOfInt(vararg deltas: Int) = listOfString().map { Integer.parseInt(it) }.mapIndexed { index, int -> int + deltas[index % deltas.size] }
fun listOfLong() = listOfString().map(String::toLong)