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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
typedef long long lint;

const lint INF = 1e15;

lint dsum(lint a, lint b){
    lint res = 0;
    while(a){res += a % b; a /= b;}
    return res;
}

int main(){
    lint n, s; cin >> n >> s;
    
    if(n == s) {cout << n + 1 << endl; return 0;}
    if(n < s) {cout << -1 << endl; return 0;}

    for(lint i=2;i*i<=n;i++) if(dsum(n, i) == s) {cout << i << endl; return 0;}

    lint ans = INF, v;
    for(lint i=1;i*i<=n-s;i++){
        if((n-s) % i == 0){
            if(v = dsum(n, (n-s)/i+1) == s) ans = min(ans, (n-s)/i+1);
            if(v = dsum(n, i+1) == s) ans = min(ans, i+1);
        }
    }

    if(ans == INF) cout << -1 << endl;
    else cout << ans << endl;
    return 0;
}