GCD

GCD (Greatest Common Divisor) or HCF (Highest Common Factor) of two numbers is the largest number that divides both of them.

Let us find the GCD of 36 and 60.

We know,

• 36 = 2 * 2 * 3 * 3

• 60 = 2 * 2 * 3 * 5

When we collect all the commen factors, ie 2 * 3 * 3, we get the GCD as 12.

For example GCD of 20 and 28 is 4 and GCD of 98 and 56 is 14.

For solution suppose a=98 & b=56

a>b so put a= a-b and b is remain same so a=98-56=42 & b= 56 . Now b>a so b=b-a and a is same

b= 56-42 = 14 & a= 42 . 42 is 3 times of 14 so HCF is 14 . likewise a=36 & b=60 ,here b>a so b = 24 & a= 36 now a>b so a= 12 & b= 24 . 12 is HCF of 36 and 60 . This concept is always satisfying.

A simple solution is to find all prime factors of both numbers, then find intersection of all factors present in both numbers.

Finally, return the product of elements in the intersection.

An efficient solution is to use Euclidean algorithm which is the main algorithm used for this purpose. The idea is, GCD of two numbers doesn’t change if smaller number is subtracted from a bigger number.

// Recursive function to return gcd of a and b
int gcd(int a, int b){    
// Everything divides 0    
if (a == 0)       
return b;    
if (b == 0)       
return a;      
// base case    
if (a == b)        
return a;      
// a is greater    
if (a > b)        
return gcd(a-b, b);    
return gcd(a, b-a);}  
// Driver program to test above function
int main(){    
int a = 98, b = 56;    
cout<<"GCD of "<<a<<" and "<<b<<" is "<<gcd(a, b);    
return 0;}

Output:

GCD of 98 and 56 is 14