Problem:
Score : 600 points

Problem Statement

Given is an integer N.
Find the minimum possible positive integer k such that (1+2+\cdots+k) is a multiple of N.
It can be proved that such a positive integer k always exists.

Constraints

1 \leq N \leq 10^{15}

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N

Output

Print the answer in a line.

Sample Input 1

11

Sample Output 1

10

1+2+\cdots+10=55 holds and 55 is indeed a multple of N=11.
There are no positive integers k \leq 9 that satisfy the condition, so the answer is k = 10.

Sample Input 2

20200920

Sample Output 2

1100144

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
using System.Threading;
using System.Runtime.CompilerServices;
using System.Diagnostics.Tracing;
static class Program{
	const int mod=(int)1e9+7;
	const double eps=1e-11;
	static void Main(){
		Sc sc=new Sc();
		var n=sc.L;
		var pf2=new Pf(n);
		var li2=new List<long>();
		pf2.Get((k,v)=>{
			if(k==2){return;}
			li2.Add((long)Pow(k,v));
		});
		int m=1<<li2.Count;
		var dp=new long[m];
		dp[0]=1;
		n*=2;
		long ans=n-1;
		for(int i = 1,j=0;i<m;i++) {
			if(1<<(j+1)==i){j++;}
			dp[i]=dp[i-(1<<j)]*li2[j];
			long c=dp[i],d=n/dp[i];
			Extgcd(c,d,out long x,out long y);
			ans=Min(ans,-Min(c*x,d*y));
		}
		Console.WriteLine(ans);
	}
	static long Extgcd(long a,long b,out long x,out long y) {
		if(b==0){x=1;y=0;return a;}
		long d=Extgcd(b,a%b,out y,out x);
		y-=a/b*x;
		return d;
	}
}
public class Pf{
	public Dictionary<long,int> hs=new Dictionary<long,int>();
	public Pf(long n){
		if(n%2==0){
			int p=0;
			while((n&1)==0){n>>=1;p++;}
			hs.Add(2,p);
		}
		var q=(int)Sqrt(n)+1;
		for(int i=3;n!=1&&i<q;i+=2){
			if(n%i==0){
				int p=0;
				while(n%i==0){n/=i;p++;}
				q=(int)Sqrt(n)+1;
				hs.Add(i,p);
			}
		}
		if(n!=1){hs.Add(n,1);}
	}
	public void Get(Action<long,int> f){
		foreach(var e in hs){f(e.Key,e.Value);}
	}
}
public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),int.Parse);}
	public int[] Ia3(int a){return Array.ConvertAll((Console.ReadLine()+" "+a.ToString()).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),long.Parse);}
	public long[] La3(int a){return Array.ConvertAll((Console.ReadLine()+" "+a.ToString()).Split(),long.Parse);}
	public double[] Da2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),double.Parse);}}
	public double[] Da3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),double.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}