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: Runtime Error)



#include "bits/stdc++.h"

#include <unordered_set>
#include <unordered_map>

//#include <iostream>//POJ
//#include <vector>
//#include <string>
//#include <iomanip>
//#include <math.h>
//#include <algorithm>
//#include <cstring>
using namespace std;
#define setc cin.tie(0);ios::sync_with_stdio(0)
#define dd(n) cout<<fixed<<setprecision(n)
#define inp(x) (cin>>x,x)
#define ind(x) (cin>>x,--x)
#define repi(i,a,b) for(int i=(a), i##_len=(b); i<i##_len; ++i)
#define rep(i,n) repi(i,0,n)
#define repi_(i,a,b) for(int i=(a), i##_len=(b); i<=i##_len; ++i)
#define rep_(i,n) repi_(i,0,n)
#define repir(i,a,b) for(int i=(a)-1, i##_first=(b); i>=i##_first; --i)
#define repr(i,n) repir(i,n,0)
#define repir_(i,a,b) for(int i=(a), i##_first=(b); i>=i##_first; --i)
#define repr_(i,n) repir_(i,n,0)
#define all(x) (x).begin(),(x).end()
#define vsort(v) sort((v).begin(), (v).end())
#define vcsort(v,c) sort((v).begin(),(v).end(),c)
#define gsort(v) tsort(v); reverse((v).begin(), (v).end())
#define vrev(v) reverse((v).begin(), (v).end())
#define fnd(v,x) ((v).find(x)!=string::npos)
#define mod(a,b) (a<0?a%b+abs(b):a%b)
#define cl(a,b) (a+b-1/b)

//template<typename T> T vpop(vector<T> *v) {
//	T res = v->back();
//	v->pop_back();
//	return res;
//}
//char vpop(string *v) {
//	char res = v->back();
//	v->pop_back();
//	return res;
//}
#define siz(v) ((int)(v).size())
#define ers(v, n) (v).erase((v).begin() + n)
#define cnt(v, n) count(all(v), n)
#define vmin(v) *min_element(v.begin(), v.end())
#define vmax(v) *max_element(v.begin(), v.end())
#define contain(q) !q.empty()
#define cont(q) !q.empty()
//#define qpop(q, a, b) a=q.back().first;b=q.back().second;q.pop()
//#define pqpop(p, a, b) a=q.top().first;b=q.top().second;q.pop()
#define el "\n"
#define sp " "
#define pi 3.14159265358979
#define co(x) cout<<x<<el
#define coc(c, a, b) if(c)co((a));else co((b))
#define cim(x) {cin>>x;--x;}
#define cim2(a,b) {cin>>a>>b;--a;--b;}
#define cosp(x) cout<<(x)<<' '
#define YES(c) coc(c,"YES", "NO")
#define Yes(c) coc(c,"Yes", "No")
#define yes(c) coc(c,"yes", "no")
#define POSSIBLE(c) coc(c, "POSSIBLE", "IMPOSSIBLE")
#define Possible(c) coc(c, "Possible", "Impossible")
#define possible(c) coc(c, "possible", "impossible")
#define inf INT_MAX
#define wildcard(T) numeric_limits<T>::min()
#define noway() {co(-1);gc();return 0;}

#define pb push_back
#define pq priority_queue
#define np next_permutation

#ifdef tqktmp_2
#define gc() getchar(); getchar()
#else
#define gc() 1
#endif
const unsigned int bf0 = (1 << 0);
const unsigned int bf1 = (1 << 1);
const unsigned int bf2 = (1 << 2);
const unsigned int bf3 = (1 << 3);
const unsigned int bf4 = (1 << 4);
const unsigned int bf5 = (1 << 5);
const unsigned int bf6 = (1 << 6);
const unsigned int bf7 = (1 << 7);
const int _10j9p7 = 1000000007;

//#define lint long long
typedef long long lint;
typedef vector<int> IV; typedef vector<string> SV;
typedef vector<lint> LIV;
typedef vector<vector<int>> IVV;
typedef pair<int, int> P; typedef pair<lint, lint> LP;
typedef vector<P> PV; typedef vector<LP> LPV;

const int
dx8[8] ={ 0,1,1,1,0,-1,-1,-1 },
dy8[8] ={ 1,1,0,-1,-1,-1,0,1 },
dx9[9] ={ 0,1,1,1,0,0,-1,-1,-1 },
dy9[9] ={ 1,1,0,-1,0,-1,-1,0,1 },
dx5[5] ={ 0,1,0,0,-1 },
dy5[5] ={ 1,0,0,-1,0 },
dx4[4] ={ 0,1,0,-1 },
dy4[4] ={ 1,0,-1,0 };
//template<class... A> void cim_(A... args) {
//	for (A *i : initializer_list<A*>{ args... }) {
//		cin>>*i;--*i;
//	}
//	return;
//}

template<class T>inline bool maxi(T &a, const T &b) { if (a < b) { a = b; return 1; } return 0; }
template<class T>inline bool mini(T &a, const T &b) { if (b < a) { a = b; return 1; } return 0; }

lint ret;
lint f(lint b, lint n){ret=0;
	while (n){
		ret+=n%b;n/=b;
	}return ret;
}

int main(){
	setc;

	lint n,s;cin>>n>>s;
	if(n==s){co(n+1);gc();return 0; }
	for (lint i=2; i*i<n; ++i){//i~b
		if (f(i, n)==s){
			co(i);gc();return 0;
		}
	}lint b;
	for (lint i=1; i*i<n; ++i){//i~p
		b=(n-s)/i+1;
		if (f(b, n)==s){
			co(i); gc(); return 0;
		}
	}co(-1);
	gc();
}