# STRGMNZ - Editorial

Author: S.Manuj Nanthan
Tester : Aryan Choudhary

Easy

# PREREQUISITES:

Maths, Observation

# PROBLEM:

You are given an integer N. Find another integer X(X>N) such that the following conditions are satisfied :

• X is not a multiple of NN.
• The value |gcd(N,X)−lcm(N,X)| is as small as possible.

If there are multiple answers for X, print any.

# EXPLANATION:

Let g = gcd(N, X) and let Y = \dfrac{X}{g}.

Now, |gcd(N, X) - lcm(N, X)| = |g - \dfrac{NX}{g}| = |g - NY|

Since X > N and X is not multiple of N, we have g < N.

So g < NY and hence |gcd(N, X) - lcm(N, X)| = |g - \dfrac{NX}{g}| = |g - NY| = NY - g

Now gcd(N, X) = g implies gcd(\dfrac{N}{g}, Y) = 1. Since X > N, we have Y > \dfrac{N}{g}.

So, Y \ge \dfrac{N}{g} + 1. But gcd(\dfrac{N}{g}, \dfrac{NX}{g} + 1) = 1. So, the minimum occurs at Y = \dfrac{N}{g} + 1.

So for fixed g = gcd(N, X), the minimum is equal to N(\dfrac{N}{g} + 1) - g = \dfrac{N^2}{g} - g + 1

The function \dfrac{N^2}{g} - g + 1 is decreasing function over g. So choose g as big as possible.

Let p be the smallest prime factor N, then choose g = \dfrac{N}{p} (largest factor of N)

And the answer will be equal to X = gY = g(\dfrac{N}{g} + 1) = N + g = N + N/p.

# TIME COMPLEXITY:

O(\sqrt{N}) per test case

# SOLUTIONS:

Author's Solution
#from itertools import *
#from math import *
#from bisect import *
#from collections import *
#from random import *
#from decimal import *
#from heapq import *
#from itertools import *            #Life is hard!
import sys
def inp():
return int(input())
def st():
return input().rstrip('\n')
def lis():
return list(map(int,input().split()))
def ma():
return map(int,input().split())
t=inp()
while(t):
t-=1
n=inp()
fl=0
for i in range(2,int(n**0.5)+1):
if(n%i==0):
fl=i
break
if(fl==0):
print(n+1)
else:
xx=(fl+1)*(n//fl)
print(xx)

Tester's Solution
/* in the name of Anton */

/*
Compete against Yourself.
Author - Aryan (@aryanc403)
Atcoder library - https://atcoder.github.io/ac-library/production/document_en/
*/

#ifdef ARYANC403
#else
#pragma GCC optimize ("Ofast")
#pragma GCC target ("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx")
//#pragma GCC optimize ("-ffloat-store")
#include<bits/stdc++.h>
#define dbg(args...) 42;
#endif

// y_combinator from @neal template https://codeforces.com/contest/1553/submission/123849801
// http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2016/p0200r0.html
template<class Fun> class y_combinator_result {
Fun fun_;
public:
template<class T> explicit y_combinator_result(T &&fun): fun_(std::forward<T>(fun)) {}
template<class ...Args> decltype(auto) operator()(Args &&...args) { return fun_(std::ref(*this), std::forward<Args>(args)...); }
};
template<class Fun> decltype(auto) y_combinator(Fun &&fun) { return y_combinator_result<std::decay_t<Fun>>(std::forward<Fun>(fun)); }

using namespace std;
#define fo(i,n)   for(i=0;i<(n);++i)
#define repA(i,j,n)   for(i=(j);i<=(n);++i)
#define repD(i,j,n)   for(i=(j);i>=(n);--i)
#define all(x) begin(x), end(x)
#define sz(x) ((lli)(x).size())
#define pb push_back
#define mp make_pair
#define X first
#define Y second
#define endl "\n"

typedef long long int lli;
typedef long double mytype;
typedef pair<lli,lli> ii;
typedef vector<ii> vii;
typedef vector<lli> vi;

const auto start_time = std::chrono::high_resolution_clock::now();
void aryanc403()
{
#ifdef ARYANC403
auto end_time = std::chrono::high_resolution_clock::now();
std::chrono::duration<double> diff = end_time-start_time;
cerr<<"Time Taken : "<<diff.count()<<"\n";
#endif
}

long long readInt(long long l, long long r, char endd) {
long long x=0;
int cnt=0;
int fi=-1;
bool is_neg=false;
while(true) {
char g=getchar();
if(g=='-') {
assert(fi==-1);
is_neg=true;
continue;
}
if('0'<=g&&g<='9') {
x*=10;
x+=g-'0';
if(cnt==0) {
fi=g-'0';
}
cnt++;
assert(fi!=0 || cnt==1);
assert(fi!=0 || is_neg==false);

assert(!(cnt>19 || ( cnt==19 && fi>1) ));
} else if(g==endd) {
if(is_neg) {
x=-x;
}
assert(l<=x&&x<=r);
return x;
} else {
assert(false);
}
}
}
string readString(int l, int r, char endd) {
string ret="";
int cnt=0;
while(true) {
char g=getchar();
assert(g!=-1);
if(g==endd) {
break;
}
cnt++;
ret+=g;
}
assert(l<=cnt&&cnt<=r);
return ret;
}
long long readIntSp(long long l, long long r) {
}
long long readIntLn(long long l, long long r) {
}
string readStringLn(int l, int r) {
}
string readStringSp(int l, int r) {
}

assert(getchar()==EOF);
}

vi readVectorInt(int n,lli l,lli r){
vi a(n);
for(int i=0;i<n-1;++i)
return a;
}

// #include<atcoder/dsu>
// vector<vi> readTree(const int n){
//     vector<vi> e(n);
//     atcoder::dsu d(n);
//     for(lli i=1;i<n;++i){
//         const lli u=readIntSp(1,n)-1;
//         const lli v=readIntLn(1,n)-1;
//         e[u].pb(v);
//         e[v].pb(u);
//         d.merge(u,v);
//     }
//     assert(d.size(0)==n);
//     return e;
// }

const lli INF = 0xFFFFFFFFFFFFFFFL;

lli seed;
inline lli rnd(lli l=0,lli r=INF)
{return uniform_int_distribution<lli>(l,r)(rng);}

class CMP
{public:
bool operator()(ii a , ii b) //For min priority_queue .
{    return ! ( a.X < b.X || ( a.X==b.X && a.Y <= b.Y ));   }};

void add( map<lli,lli> &m, lli x,lli cnt=1)
{
auto jt=m.find(x);
if(jt==m.end())         m.insert({x,cnt});
else                    jt->Y+=cnt;
}

void del( map<lli,lli> &m, lli x,lli cnt=1)
{
auto jt=m.find(x);
if(jt->Y<=cnt)            m.erase(jt);
else                      jt->Y-=cnt;
}

bool cmp(const ii &a,const ii &b)
{
return a.X<b.X||(a.X==b.X&&a.Y<b.Y);
}

const lli mod = 1000000007L;
// const lli maxN = 1000000007L;

lli T,n,i,j,k,in,cnt,l,r,u,v,x,y;
lli m;
string s;
vi a;
//priority_queue < ii , vector < ii > , CMP > pq;// min priority_queue .

int main(void) {
ios_base::sync_with_stdio(false);cin.tie(NULL);
// freopen("txt.in", "r", stdin);
// freopen("txt.out", "w", stdout);
// cout<<std::fixed<<std::setprecision(35);
const auto solve=[&](lli n)->lli{
if(n==1)
return -1;
for(lli i=2;i*i<=n;++i){
if(n%i)
continue;
return (i+1)*(n/i);
}
return (n+1);
};