Check if the given two numbers are friendly pair or not

Given two positive integer N, M. The task is to check if N and M are friendly pair or not.

In number theory, friendly pairs are two numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself i.e ?(n)/n. S o, two number n and m are friendly number if
?(n)/n = ?(m)/m.
where ?(n) is the sum of divisors of n.

Examples:



Input : n = 6, m = 28
Output : Yes
Explanation:
Divisor of 6 are 1, 2, 3, 6.
Divisor of 28 are 1, 2, 4, 7, 14, 28.
Sum of divisor of 6 and 28 are 12 and 56
respectively. Abundancy index of 6 and 28 
are 2. So they are friendly pair.

Input : n = 18, m = 26
Output : No

The idea is to find the sum of divisor of n and m. And to check if the abundancy index of n and m, we will find the Greatest Common Divisors of n and m with their sum of divisors. And check if the reduced form of abundancy index of n and m are equal by checking if their numerator and denominator are equal or not. To find the reduced form, we will divide numerator and denominator by GCD.

Below is the implementation of above idea :

C++

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// Check if the given two number 
// are friendly pair or not.
#include <bits/stdc++.h>
using namespace std;
  
// Returns sum of all factors of n.
int sumofFactors(int n)
{
  
    // Traversing through all prime factors.
    int res = 1;
    for (int i = 2; i <= sqrt(n); i++) {
  
        int count = 0, curr_sum = 1;
        int curr_term = 1;
        while (n % i == 0) {
            count++;
  
            // THE BELOW STATEMENT MAKES
            // IT BETTER THAN ABOVE METHOD
            // AS WE REDUCE VALUE OF n.
            n = n / i;
  
            curr_term *= i;
            curr_sum += curr_term;
        }
  
        res *= curr_sum;
    }
  
    // This condition is to handle
    // the case when n is a prime
    // number greater than 2.
    if (n >= 2)
        res *= (1 + n);
  
    return res;
}
  
// Function to return gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
  
// Function to check if the given two 
// number are friendly pair or not.
bool checkFriendly(int n, int m)
{
    // Finding the sum of factors of n and m
    int sumFactors_n = sumofFactors(n);
    int sumFactors_m = sumofFactors(m);
  
    // finding gcd of n and sum of its factors.
    int gcd_n = gcd(n, sumFactors_n);
  
    // findig gcd of m and sum of its factors.
    int gcd_m = gcd(m, sumFactors_m);
  
    // checking is numerator and denominator of 
    // abundancy index of both number are equal
    // or not.
    if (n / gcd_n == m / gcd_m && 
        sumFactors_n / gcd_n == sumFactors_m / gcd_m)
        return true;
  
    else
        return false;
}
  
// Driver code
int main()
{
    int n = 6, m = 28;
    checkFriendly(n, m) ? (cout << "Yes\n") : 
                          (cout << "No\n");
    return 0;
}

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Java

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// Java code to check if the given two number 
// are friendly pair or not.
class GFG {
      
    // Returns sum of all factors of n.
    static int sumofFactors(int n)
    {
      
        // Traversing through all prime factors.
        int res = 1;
        for (int i = 2; i <= Math.sqrt(n); i++) {
      
            int count = 0, curr_sum = 1;
            int curr_term = 1;
            while (n % i == 0) {
                count++;
      
                // THE BELOW STATEMENT MAKES
                // IT BETTER THAN ABOVE METHOD
                // AS WE REDUCE VALUE OF n.
                n = n / i;
      
                curr_term *= i;
                curr_sum += curr_term;
            }
      
            res *= curr_sum;
        }
      
        // This condition is to handle
        // the case when n is a prime
        // number greater than 2.
        if (n >= 2)
            res *= (1 + n);
      
        return res;
    }
      
    // Function to return gcd of a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
              
        return gcd(b % a, a);
    }
      
    // Function to check if the given two 
    // number are friendly pair or not.
    static boolean checkFriendly(int n, int m)
    {
        // Finding the sum of factors of n and m
        int sumFactors_n = sumofFactors(n);
        int sumFactors_m = sumofFactors(m);
      
        // finding gcd of n and sum of its factors.
        int gcd_n = gcd(n, sumFactors_n);
      
        // findig gcd of m and sum of its factors.
        int gcd_m = gcd(m, sumFactors_m);
      
        // checking is numerator and denominator of 
        // abundancy index of both number are equal
        // or not.
        if (n / gcd_n == m / gcd_m && 
            sumFactors_n / gcd_n == sumFactors_m / gcd_m)
            return true;
      
        else
            return false;
    }
      
    //driver code
    public static void main (String[] args)
    {
        int n = 6, m = 28;
          
        if(checkFriendly(n, m))
            System.out.print("Yes\n");
        else
            System.out.print("No\n");
    }
}
  
// This code is contributed by Anant Agarwal.

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Python3

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# Check if the given two number 
# are friendly pair or not.
import math
  
# Returns sum of all factors of n.
def sumofFactors(n):
  
    # Traversing through all prime factors.
    res = 1
    for i in range(2, int(math.sqrt(n)) + 1):
  
        count = 0; curr_sum = 1; curr_term = 1
        while (n % i == 0):
            count += 1
  
            # THE BELOW STATEMENT MAKES
            # IT BETTER THAN ABOVE METHOD
            # AS WE REDUCE VALUE OF n.
            n = n // i
  
            curr_term *= i
            curr_sum += curr_term
          
        res *= curr_sum
      
    # This condition is to handle
    # the case when n is a prime
    # number greater than 2.
    if (n >= 2):
        res *= (1 + n)
  
    return res
  
# Function to return gcd of a and b
def gcd(a, b):
  
    if (a == 0):
        return b
    return gcd(b % a, a)
  
# Function to check if the given two 
# number are friendly pair or not.
def checkFriendly(n, m):
  
    # Finding the sum of factors of n and m
    sumFactors_n = sumofFactors(n)
    sumFactors_m = sumofFactors(m)
  
    # Finding gcd of n and sum of its factors.
    gcd_n = gcd(n, sumFactors_n)
  
    # Findig gcd of m and sum of its factors.
    gcd_m = gcd(m, sumFactors_m)
  
    # checking is numerator and denominator  
    # of abundancy index of both number are 
    # equal or not.
    if (n // gcd_n == m // gcd_m and
        sumFactors_n // gcd_n == sumFactors_m // gcd_m):
        return True
  
    else:
        return False
  
# Driver code
n = 6; m = 28
if(checkFriendly(n, m)):
    print("Yes")
else:
    print("No")
      
# This code is contributed by Anant Agarwal.

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C#

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// C# code to check if the
// given two number are 
// friendly  pair or not
using System;
  
class GFG {
      
    // Returns sum of all 
    //factors of n.
    static int sumofFactors(int n)
    {
      
        // Traversing through all 
        // prime factors.
        int res = 1;
        for (int i = 2; i <= Math.Sqrt(n); i++) 
        {
      
            int count = 0, curr_sum = 1;
            int curr_term = 1;
            while (n % i == 0) 
            {
                count++;
      
                // THE BELOW STATEMENT MAKES
                // IT BETTER THAN ABOVE METHOD
                // AS WE REDUCE VALUE OF n.
                n = n / i;
      
                curr_term *= i;
                curr_sum += curr_term;
            }
      
            res *= curr_sum;
        }
      
        // This condition is to handle
        // the case when n is a prime
        // number greater than 2.
        if (n >= 2)
            res *= (1 + n);
      
        return res;
    }
      
    // Function to return gcd 
    // of a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
              
        return gcd(b % a, a);
    }
      
    // Function to check if the 
    // given two number are 
    // friendly pair or not
    static bool checkFriendly(int n, int m)
    {
        // Finding the sum of factors 
        // of n and m
        int sumFactors_n = sumofFactors(n);
        int sumFactors_m = sumofFactors(m);
      
        // finding gcd of n and 
        // sum of its factors.
        int gcd_n = gcd(n, sumFactors_n);
      
        // findig gcd of m and sum
        // of its factors.
        int gcd_m = gcd(m, sumFactors_m);
      
        // checking is numerator and 
        // denominator of abundancy
        // index of both number are
        // equal or not
        if (n / gcd_n == m / gcd_m && 
            sumFactors_n / gcd_n == 
            sumFactors_m / gcd_m)
            return true;
      
        else
            return false;
    }
      
    // Driver code
    public static void Main (String[] args)
    {
        int n = 6, m = 28;
          
        if(checkFriendly(n, m))
            Console.Write("Yes\n");
        else
            Console.Write("No\n");
    }
}
  
// This code is contributed by parshar...

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PHP

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<?php
// PHP program to check if the given
// two number are friendly pair or not.
  
// Returns sum of all factors of n.
function sumofFactors( $n)
{
  
    // Traversing through all
    // prime factors.
    $res = 1;
    for ($i = 2; $i <= sqrt($n); $i++) 
    {
  
        $count = 0;
        $curr_sum = 1;
        $curr_term = 1;
        while ($n % $i == 0)
        {
            $count++;
  
            // THE BELOW STATEMENT MAKES
            // IT BETTER THAN ABOVE METHOD
            // AS WE REDUCE VALUE OF n.
            $n = $n / $i;
  
            $curr_term *= $i;
            $curr_sum += $curr_term;
        }
  
        $res *= $curr_sum;
    }
  
    // This condition is to handle
    // the case when n is a prime
    // number greater than 2.
    if ($n >= 2)
        $res *= (1 + $n);
  
    return $res;
}
  
// Function to return 
// gcd of a and b
function gcd($a, $b)
{
    if ($a == 0)
        return $b;
    return gcd($b % $a, $a);
}
  
// Function to check if the given two 
// number are friendly pair or not.
function checkFriendly($n, $m)
{
      
    // Finding the sum of
    // factors of n and m
    $sumFactors_n = sumofFactors($n);
    $sumFactors_m = sumofFactors($m);
  
    // finding gcd of n and 
    // sum of its factors.
    $gcd_n = gcd($n, $sumFactors_n);
  
    // findig gcd of m and
    // sum of its factors.
    $gcd_m = gcd($m, $sumFactors_m);
  
    // checking is numerator 
    // and denominator of 
    // abundancy index of 
    // both number are equal
    // or not.
    if ($n / $gcd_n == $m / $gcd_m and
        $sumFactors_n / $gcd_n == 
        $sumFactors_m / $gcd_m)
        return true;
  
    else
        return false;
}
  
    // Driver code
    $n = 6;
    $m = 28;
    if(checkFriendly($n, $m))
        echo "Yes"
    else
        echo "No";
  
// This code is contributed by anuj_67.
?>

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Output

Yes


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