Find two proper factors of N such that their sum is coprime with N

Given an integer N, you have to find two proper factors of N such that their sum is coprime with the given integer N. If no such factors exist, print -1.

Examples:

Input: N = 15
Output: 3, 5
Explanation: 3 and 5 are the proper factors of 15 and 3+5 -> 8 is coprime with 15.

Input: N = 4
Output: -1
Explanation: there are no proper factors that satisfy the required conditions

Naive Approach: Generate a list of all the proper factors of N and for each possible pair, check if their sum is coprime with N i.e. GCD(sum of pair of integers, N) = 1. Here GCD means Greatest Common Divisor.

Efficient Approach:  If two numbers A and B are coprime then their sum is coprime with their product. Keeping that in mind, find all the factors of N and for each factor d1, calculate the largest factor of N, d2 that is coprime with d1. To calculate d2, simply divide N with d1 until N%d1 != 0. Finally, check if d1 and d2 are proper factors of N or not (i.e., d1>1 and d2>1).

Below is the implementation of the above approach:

C++

// C++ Program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find two proper
// factors of N such that their
// sum is coprime with N
void printFactors(int n)
{

    // Find factors in sorted order
    for (int i = 2; i <= sqrt(n); i++) {

        if (n % i == 0) {
            int d1 = i, d2 = n;

            // Find largest value of d2 such
            // that d1 and d2 are co-prime
            while (d2 % d1 == 0) {
                d2 = d2 / d1;
            }

            // Check if d1 and d2 are proper
            // factors of N
            if (d1 > 1 && d2 > 1) {
                // Print answer
                cout << d1 << ", " << d2;
                return;
            }
        }
    }

    // No such factors exist
    cout << -1;
}

// Driver code
int main()
{
    int N = 10;

    // Function Call
    printFactors(N);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG{

// Function to find two proper
// factors of N such that their
// sum is coprime with N
static void printFactors(int n)
{

    // Find factors in sorted order
    for(int i = 2; i <= (int)Math.sqrt(n); i++)
    {
        if (n % i == 0)
        {
            int d1 = i, d2 = n;

            // Find largest value of d2 such
            // that d1 and d2 are co-prime
            while (d2 % d1 == 0)
            {
                d2 = d2 / d1;
            }

            // Check if d1 and d2 are proper
            // factors of N
            if (d1 > 1 && d2 > 1)
            {

                // Print answer
                System.out.print(d1 + ", " + d2);
                return;
            }
        }
    }

    // No such factors exist
    System.out.print(-1);
}

// Driver code
public static void main(String[] args)
{
    int N = 10;

    // Function Call
    printFactors(N);
}
}

// This code is contributed by Potta Lokesh

Python3

# Python Program for the above approach
import math

# Function to find two proper
# factors of N such that their
# sum is coprime with N
def printFactors(n):
    # Find factors in sorted order
    for i in range(2, int(math.sqrt(n))+1):
        if (n % i == 0):
            d1 = i
            d2 = n

            # Find largest value of d2 such
            # that d1 and d2 are co-prime
            while (d2 % d1 == 0):
                d2 = d2 // d1

            # Check if d1 and d2 are proper
            # factors of N
            if (d1 > 1 and d2 > 1):

                # Print answer
                print(d1, d2, sep=", ")
                return
    # No such factors exist
    print(-1)
# Driver code
N = 10

# Function Call
printFactors(N)

# This code is contributed by Shivani

C#

// C# Program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find two proper
// factors of N such that their
// sum is coprime with N
static void printFactors(int n)
{

    // Find factors in sorted order
    for (int i = 2; i <= (int)Math.Sqrt(n); i++) {

        if (n % i == 0) {
            int d1 = i, d2 = n;

            // Find largest value of d2 such
            // that d1 and d2 are co-prime
            while (d2 % d1 == 0) {
                d2 = d2 / d1;
            }

            // Check if d1 and d2 are proper
            // factors of N
            if (d1 > 1 && d2 > 1)
            {

                // Print answer
                Console.Write(d1 + ", "+d2);
                return;
            }
        }
    }

    // No such factors exist
    Console.Write(-1);
}

// Driver code
public static void Main()
{
    int N = 10;

    // Function Call
    printFactors(N);
}
}

// This code is contributed by ipg2016107.

Javascript

<script>
// Javascript Program for the above approach

// Function to find two proper
// factors of N such that their
// sum is coprime with N
function printFactors(n) {
  // Find factors in sorted order
  for (let i = 2; i <= Math.sqrt(n); i++) {
    if (n % i == 0) {
      let d1 = i,
        d2 = n;

      // Find largest value of d2 such
      // that d1 and d2 are co-prime
      while (d2 % d1 == 0) {
        d2 = Math.floor(d2 / d1);
      }

      // Check if d1 and d2 are proper
      // factors of N
      if (d1 > 1 && d2 > 1) {
        // Print answer
        document.write(d1 + ", " + d2);
        return;
      }
    }
  }

  // No such factors exist
  document.write(-1);
}

// Driver code

let N = 10;

// Function Call
printFactors(N);

// This code is contributed by _saurabh_jaiswal.
</script>

Output:

2, 5

Time Complexity: O(√N)
Auxiliary Space: O(1)