Here’s how the Euclidean algorithm works step-by-step:

1. Input Two Numbers: Start with two positive integers, \(a\) and \(b\), where \(a \geq b\).

2. Divide and Find Remainder: Divide \(a\) by \(b\) and find the remainder, \(r\). Mathematically, this can be expressed as \(a = bq + r\), where \(q\) is the quotient.

3. Update the Numbers: Replace \(a\) with \(b\) and \(b\) with \(r\).

4. Repeat: Continue the process of dividing, finding the remainder, and updating the numbers until the remainder is zero.

5. Conclusion: When the remainder reaches zero, the last non-zero remainder is the greatest common divisor of the original two numbers.

Example

To find the GCD of 252 and 198:

- \(252 = 198 \times 1 + 54\) (Here, \(a=252\), \(b=198\), remainder \(r=54\))

- \(198 = 54 \times 3 + 36\) (Now, \(a=198\), \(b=54\), new remainder \(r=36\))

- \(54 = 36 \times 1 + 18\) (Now, \(a=54\), \(b=36\), new remainder \(r=18\))

- \(36 = 18 \times 2 + 0\) (Finally, \(a=36\), \(b=18\), remainder \(r=0\))

The GCD of 252 and 198 is 18, as that is the last non-zero remainder.

Properties

- Efficiency: The Euclidean algorithm is very efficient and works well even for very large numbers, making it practical for computational applications.

- Extension: This algorithm can be extended to not just numbers but also to polynomials and other mathematical elements.

The Euclidean algorithm is fundamental in number theory and has many practical applications, including simplifying fractions, computing least common multiples, and in cryptographic algorithms like RSA.