# PROBLEM LINK:

https://www.codechef.com/START24C/problems/DIVREC

Setter: Harsh Vardhan Goenka

Tester: Aryan Chaudhary

Editorialist: Rishabh Gupta

# DIFFICULTY:

Easy

# PREREQUISITES:

None

# PROBLEM:

Alice is teaching Bob maths via a game called N-guesser.

Alice has a number N which Bob needs to guess.

Alice gives Bob a number X which represents the sum of divisors of N. Alice further provides Bob with A and B where A/B represents the sum of reciprocals of divisors of N.

Bob either needs to guess number N or tell that no such number exists.

# EXPLANATION:

The divisors of a number n are of the form a_1, n/a_1, a_2, n/a_2, a_3, n/a_3 â€¦sqrt(n) (only if it divides n). So, the sum of reciprocal of the divisors of a number is nothing but the sum of divisors of the number divided by the number n itself.

So, we can the get the number n as X*B/A. But, we have to check that this n yields the same sum of the divisors as given input X, which will require O(sqrt(n)) time.

# TIME COMPLEXITY:

O(sqrt(n)) for each test case.

# SOLUTION:

## Setter's Solution

```
#include <bits/stdc++.h>
using namespace std;
int main(){
long long t;
cin>>t;
while(t--){
long long n,x,a,b;
cin>>x>>a>>b;
long long c=gcd(a,b);
while(c!=1){
c=c;
}
n=(x*b);
if(n%a!=0){
cout<<"-1\n";
continue;
}
n/=a;
if(n>1e9){
cout<<"-1\n";
continue;
}
long long sum=0ll;
for(long long i=1;i*i<=n;i++){
if(n%i==0){
sum+=i;
if(i!=n/i)
sum+=n/i;
}
}
if(sum==x)
cout<<n<<"\n";
else
cout<<"-1\n";
}
}
```

## Editorialist''s Solution

```
#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define dd double
#define endl "\n"
#define pb push_back
#define all(v) v.begin(),v.end()
#define mp make_pair
#define fi first
#define se second
#define vll vector<ll>
#define pll pair<ll,ll>
#define fo(i,n) for(int i=0;i<n;i++)
#define fo1(i,n) for(int i=1;i<=n;i++)
ll mod=1000000007;
ll n,k,t,m,q,flag=0;
ll power(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
// #include <ext/pb_ds/assoc_container.hpp>
// #include <ext/pb_ds/tree_policy.hpp>
// using namespace __gnu_pbds;
// #define ordered_set tree<int, null_type,less<int>, rb_tree_tag,tree_order_statistics_node_update>
// ordered_set s ; s.order_of_key(a) -- no. of elements strictly less than a
// s.find_by_order(i) -- itertor to ith element (0 indexed)
ll min(ll a,ll b){if(a>b)return b;else return a;}
ll max(ll a,ll b){if(a>b)return a;else return b;}
ll gcd(ll a , ll b){ if(b > a) return gcd(b , a) ; if(b == 0) return a ; return gcd(b , a%b) ;}
int main() {
ios_base::sync_with_stdio(0);
cin.tie(0); cout.tie(0);
#ifdef NOOBxCODER
freopen("input.txt", "r", stdin);
freopen("output.txt", "w", stdout);
#else
#define NOOBxCODER 0
#endif
cin>>t;
//t=1;
while(t--){
ll a,b,c;
cin>>a>>b>>c;
if(a%b !=0 || c>b){cout<<-1<<endl; continue;}
c *= a/b;
ll sum=0;
for(int i=1;i*i<=c; i++){
if(c%i!=0)continue;
if(i*i==c)sum+=i;
else sum+= i + c/i;
}
if(sum == a)cout<<c<<endl;
else cout<<-1<<endl;
}
cerr << "Time : " << 1000 * ((double)clock()) / (double)CLOCKS_PER_SEC << "ms\n";
return 0;
}
```