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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

using namespace std;
typedef long long ll;
typedef pair<int,int>pa;

const int N=2e5+100;

ll A(ll x,ll y)
{
    //if(y==1)
    if(x<y)
        return x;
    return A(x/y,y)+x%y;
}
int main()
{
    ios::sync_with_stdio(0);cin.tie(0);
    ll n,s;
    cin>>n>>s;
    if(n==s) {
        cout<<n+1<<endl;
        return 0;
    }
    if(n<s){
        cout<<-1<<endl;
        return 0;
    }
    for(ll i=2;i*i<=n+1;i++){
       if(A(n,i)==s) {
        cout<<i;
        return 0;
       }
    }
    ll m=n-s;
    ll ans=1e12;
    for(ll i=1;i*i<=m;i++){
        if(m%i==0){
             if(A(n,m/i+1)==s)
             ans=min(ans,m/i+1);
        }
    }
    if(ans==1e12)
    cout<<-1<<endl;
    else cout<<ans<<endl;
    return 0;
}