Check if the given two numbers are friendly pair or not

23 Jun, 2022
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++
Java
Python3
C#
PHP
Javascript
// 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 bint 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);     // finding 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 codeint main(){    int n = 6, m = 28;    checkFriendly(n, m) ? (cout << "Yes\n") :                          (cout << "No\n");    return 0;}
// 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);             // finding 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.
# 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 bdef 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)     # Finding 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 coden = 6; m = 28if(checkFriendly(n, m)):    print("Yes")else:    print("No")     # This code is contributed by Anant Agarwal.
// C# code to check if the// given two number are// friendly  pair or notusing 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);             // finding 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...
<?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 $res;} // Function to return// gcd of a and bfunction gcd($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);     // finding 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.?>
<script> // JavaScript 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.        let res = 1;        for (let i = 2; i <= Math.sqrt(n); i++) {                   let count = 0, curr_sum = 1;            let 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        let sumFactors_n = sumofFactors(n);        let sumFactors_m = sumofFactors(m);               // finding gcd of n and sum of its factors.        let gcd_n = gcd(n, sumFactors_n);               // finding gcd of m and sum of its factors.        let 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         let n = 6, m = 28;        if(checkFriendly(n, m))            document.write("Yes\n");        else            document.write("No\n"); // This code is contributed by avijitmondal1998.</script>
Output  
Yes
Time Complexity: O(√n log n) Auxiliary Space: O(1)
Please suggest if someone has a better solution which is more efficient in terms of space and time.This article is contributed by Aarti_Rathi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
parashar
vt_m
nidhi_biet
avijitmondal1998
sumitgumber28
codewithmini
divisors
GCD-LCM
number-theory
Mathematical
number-theory
Mathematical
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Algorithm to solve Rubik's Cube
Merge two sorted arrays with O(1) extra space
Program to print prime numbers from 1 to N.
Segment Tree | Set 1 (Sum of given range)
Check if a number is Palindrome
Count ways to reach the n'th stair
Fizz Buzz Implementation
Median of two sorted arrays of same size
Sum of the series (1*2) + (2*3) + (3*4) + ...... upto n terms
Product of Array except itself