Using Euclid's Algorithm to Find the Greatest Common Divisor of Two Positive Integers

Using Euclid’s Algorithm to Find the Greatest Common Divisor of Two Positive Integers

Understanding Euclid’s Algorithm

Euclid’s Algorithm is a process used to find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of two positive integers.
It is an iterative method which reduces the problem of finding the GCD of two numbers to the problem of finding the GCD of smaller pairs of numbers.
The algorithm is based on the principle that the GCD of two numbers also divides their difference.

Applying Euclid’s Algorithm Step by Step

Step 1: Arrange the two integers in decreasing order. The greater integer becomes ‘a’ and the lesser integer is termed ‘b’.

Step 2: Divide ‘a’ by ‘b’. Calculate the quotient and the remainder.

Step 3: Replace ‘a’ with ‘b’ and ‘b’ with the remainder from Step 2.

Step 4: Repeat Step 2 and Step 3 until you get a remainder of zero.

Step 5: The last non-zero remainder obtained in the process is the GCD of the original pair of numbers.

Example of Euclid’s Algorithm

Consider two positive integers, 420 and 120.

Firstly, divide the larger number (420) by the smaller number (120). The quotient is 3 and the remainder is 60.
Now, replace 420 with 120 and 120 with the remainder 60, and carry out the division again. The new quotient is 2 and the remainder is 0.
Since the remainder is now zero, the process stops. The last non-zero remainder (which is 60) is the GCD of 420 and 120.

Importance of Euclid’s Algorithm

Euclid’s Algorithm is one of the oldest algorithms in common use. It originates from Euclid’s Elements (circa 300 BC).
It is a simple and efficient method. The process requires only basic arithmetic operations, making it easy to understand and implement.
It plays a central role in number theory and is the underlying principle of several modern algorithms in cryptography and computer science.