Maximum count of common divisors of A and B such that all are co-primes to one another

Given two integers A and B. The task is to find the count of maximum elements from the common divisors of A and B such that all selected elements are co-prime to one another.

Examples:

Input: A = 12, B = 18
Output: 3
Common divisors of A and B are 1, 2, 3 and 6.
Select 1, 2, and 3. All the pairs are co primes to
one another i.e. gcd(1, 2) = gcd(1, 3) = gcd(2, 3) = 1.

Input: A = 1, B = 3
Output: 1

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: It can be observed that all the common factors of A and B must be a factor of their gcd. And, in order for the factors of this gcd to be co-prime to one another, one element of the pair must be either 1 or both the elements must be prime. So the answer will be 1 more than the count of prime divisors of gcd(A, B). Note that 1 is added because 1 can also be a part of the chosen divisors as its gcd with the other pairs will always be 1.

Below is the implementation of the above approach:

C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the count of common factors ` `// of a and b such that all the elements ` `// are co-prime to one another ` `int` `maxCommonFactors(``int` `a, ``int` `b) ` `{ ` `    ``// GCD of a and b ` `    ``int` `gcd = __gcd(a, b); ` ` `  `    ``// Include 1 initially ` `    ``int` `ans = 1; ` ` `  `    ``// Find all the prime factors of the gcd ` `    ``for` `(``int` `i = 2; i * i <= gcd; i++) { ` `        ``if` `(gcd % i == 0) { ` `            ``ans++; ` `            ``while` `(gcd % i == 0) ` `                ``gcd /= i; ` `        ``} ` `    ``} ` ` `  `    ``// If gcd is prime ` `    ``if` `(gcd != 1) ` `        ``ans++; ` ` `  `    ``// Return the required answer ` `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `a = 12, b = 18; ` ` `  `    ``cout << maxCommonFactors(a, b); ` ` `  `    ``return` `0; ` `} `

Java

 `// Java implementation of the approach  ` `class` `GFG ` `{ ` `     `  `static` `int` `gcd(``int` `a, ``int` `b)  ` `{  ` `    ``if` `(b == ``0``)  ` `        ``return` `a;  ` `    ``return` `gcd(b, a % b);  ` `}  ` ` `  `// Function to return the count of common factors  ` `// of a and b such that all the elements  ` `// are co-prime to one another  ` `static` `int` `maxCommonFactors(``int` `a, ``int` `b)  ` `{  ` `     `  `    ``// GCD of a and b  ` `    ``int` `__gcd = gcd(a, b);  ` ` `  `    ``// Include 1 initially  ` `    ``int` `ans = ``1``;  ` ` `  `    ``// Find all the prime factors of the gcd  ` `    ``for` `(``int` `i = ``2``; i * i <= __gcd; i++) ` `    ``{  ` `        ``if` `(__gcd % i == ``0``) ` `        ``{  ` `            ``ans++;  ` `            ``while` `(__gcd % i == ``0``)  ` `                ``__gcd /= i;  ` `        ``}  ` `    ``}  ` ` `  `    ``// If gcd is prime  ` `    ``if` `(__gcd != ``1``)  ` `        ``ans++;  ` ` `  `    ``// Return the required answer  ` `    ``return` `ans;  ` `}  ` ` `  `// Driver code  ` `public` `static` `void` `main (String[] args)  ` `{  ` `    ``int` `a = ``12``, b = ``18``;  ` ` `  `    ``System.out.println(maxCommonFactors(a, b));  ` `}  ` `} ` ` `  `// This code is contributed by AnkitRai01 `

Python3

 `# Python3 implementation of the approach  ` `import` `math ` ` `  `# Function to return the count of common factors  ` `# of a and b such that all the elements  ` `# are co-prime to one another  ` `def` `maxCommonFactors(a, b):  ` `     `  `    ``# GCD of a and b  ` `    ``gcd ``=` `math.gcd(a, b)  ` ` `  `    ``# Include 1 initially  ` `    ``ans ``=` `1``;  ` ` `  `    ``# Find all the prime factors of the gcd ` `    ``i ``=` `2` `    ``while` `(i ``*` `i <``=` `gcd):  ` `        ``if` `(gcd ``%` `i ``=``=` `0``):  ` `            ``ans ``+``=` `1` `            ``while` `(gcd ``%` `i ``=``=` `0``):  ` `                ``gcd ``=` `gcd ``/``/` `i ` `        ``i ``+``=` `1`     `                 `  `    ``# If gcd is prime  ` `    ``if` `(gcd !``=` `1``):  ` `        ``ans ``+``=` `1` ` `  `    ``# Return the required answer  ` `    ``return` `ans ` `     `  `# Driver code  ` `a ``=` `12` `b ``=` `18` `print``(maxCommonFactors(a, b))  ` ` `  `# This code is contributed by ` `# divyamohan123 `

C#

 `// C# implementation of the above approach  ` `using` `System;          ` ` `  `class` `GFG  ` `{ ` `     `  `    ``static` `int` `gcd(``int` `a, ``int` `b)  ` `    ``{  ` `        ``if` `(b == 0)  ` `            ``return` `a;  ` `        ``return` `gcd(b, a % b);  ` `    ``}  ` ` `  `    ``// Function to return the count of common factors  ` `    ``// of a and b such that all the elements  ` `    ``// are co-prime to one another  ` `    ``static` `int` `maxCommonFactors(``int` `a, ``int` `b)  ` `    ``{  ` ` `  `        ``// GCD of a and b  ` `        ``int` `__gcd = gcd(a, b);  ` ` `  `        ``// Include 1 initially  ` `        ``int` `ans = 1;  ` ` `  `        ``// Find all the prime factors of the gcd  ` `        ``for` `(``int` `i = 2; i * i <= __gcd; i++) ` `        ``{  ` `            ``if` `(__gcd % i == 0) ` `            ``{  ` `                ``ans++;  ` `                ``while` `(__gcd % i == 0)  ` `                    ``__gcd /= i;  ` `            ``}  ` `        ``}  ` ` `  `        ``// If gcd is prime  ` `        ``if` `(__gcd != 1)  ` `            ``ans++;  ` ` `  `        ``// Return the required answer  ` `        ``return` `ans;  ` `    ``}  ` ` `  `    ``// Driver code  ` `    ``public` `static` `void` `Main (String[] args)  ` `    ``{  ` `        ``int` `a = 12, b = 18;  ` ` `  `        ``Console.WriteLine(maxCommonFactors(a, b));  ` `    ``}  ` `} ` ` `  `// This code is contributed by Rajput-Ji `

Output:

```3
```

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