> For the complete documentation index, see [llms.txt](https://soveet-nayak.gitbook.io/algorithms/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://soveet-nayak.gitbook.io/algorithms/number-theory/gcd.md).

# 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. 

```cpp
// 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:**&#x20;

```
GCD of 98 and 56 is 14
```