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 5.4.1) (judge verdict: Accepted)

#include<stdio.h>
#include<stdlib.h>
#include<math.h>

typedef long long int int64;

#define MAX(a,b) ((a)>(b)?(a):(b))
#define MIN(a,b) ((a)<(b)?(a):(b))
#define ABS(a) ((a)>(0)?(a):-(a))

int64 f(int64 n,int b){
  int64 res=0;
  while(n>0){
    res+=n%b;
    n/=b;
  }
  return res;
}

void run(void){
  int64 n,s;
  scanf("%lld%lld",&n,&s);
  int b;
  for(b=2;(int64)b*b<=n;b++){
    if(f(n,b)==s){
      printf("%d\n",b);
      return;
    }
  }
  //n=xp+y
  int64 x;
  for(x=n/b;x>=1;x--){
    int64 y=s-x;
    if(!(y>=0 && n-y>0 && (n-y)%x==0)) continue;
    int64 p=(n-y)/x;
    if(x<p && y<p && n==x*p+y){
      printf("%lld\n",p);
      return;
    }
  }
  if(n==s){
    printf("%lld\n",n+1);
  } else {
    printf("-1\n");
  }
}

int main(void){
  run();
  return 0;
}