In this example, you will learn to find the GCD of two numbers using two different methods: function and loops and, Euclidean algorithm

To understand this example, you should have the knowledge of the following Python programming topics:

The highest common factor (H.C.F) or greatest common divisor (G.C.D) of two numbers is the largest positive integer that perfectly divides the two given numbers. For example, the H.C.F of 12 and 14 is 2.

```
# Python program to find H.C.F of two numbers
# define a function
def compute_hcf(x, y):
# choose the smaller number
if x > y:
smaller = y
else:
smaller = x
for i in range(1, smaller+1):
if((x % i == 0) and (y % i == 0)):
hcf = i
return hcf
num1 = 54
num2 = 24
print("The H.C.F. is", compute_hcf(num1, num2))
```

**Output**

The H.C.F. is 6

Here, two integers stored in variables `num1` and `num2` are passed to the `compute_hcf()`

function. The function computes the H.C.F. these two numbers and returns it.

In the function, we first determine the smaller of the two numbers since the H.C.F can only be less than or equal to the smallest number. We then use a `for`

loop to go from 1 to that number.

In each iteration, we check if our number perfectly divides both the input numbers. If so, we store the number as H.C.F. At the completion of the loop, we end up with the largest number that perfectly divides both the numbers.

The above method is easy to understand and implement but not efficient. A much more efficient method to find the H.C.F. is the Euclidean algorithm.

This algorithm is based on the fact that H.C.F. of two numbers divides their difference as well.

In this algorithm, we divide the greater by smaller and take the remainder. Now, divide the smaller by this remainder. Repeat until the remainder is 0.

For example, if we want to find the H.C.F. of 54 and 24, we divide 54 by 24. The remainder is 6. Now, we divide 24 by 6 and the remainder is 0. Hence, 6 is the required H.C.F.

```
# Function to find HCF the Using Euclidian algorithm
def compute_hcf(x, y):
while(y):
x, y = y, x % y
return x
hcf = compute_hcf(300, 400)
print("The HCF is", hcf)
```

Here we loop until `y` becomes zero. The statement `x, y = y, x % y`

does swapping of values in Python. Click here to learn more about swapping variables in Python.

In each iteration, we place the value of `y` in `x` and the remainder `(x % y)`

in `y`, simultaneously. When `y` becomes zero, we have H.C.F. in `x`.