Euclid's division lemma

Euclid's Division Algorithm

Euclid's Division Algorithm is an efficient method for finding the greatest common divisor (GCD) of two integers. It works on the idea that the greatest common divisor of two numbers also divides the remainder when the larger is divided by the smaller. This algorithm is one of the oldest and most fundamental algorithms in number theory.

What is Euclid's Division Algorithm?

\(a = bq + r \quad \; where 0 \leq r < b\):

\[ a = bq + r, quad 0 leq r < b \]

The algorithm applies this division repeatedly by replacing \(a\) with \(b\) and \(b\) with \(r\) until the remainder becomes zero. The divisor at this stage is the GCD of the original two numbers.

Steps to Use Euclid's Division Algorithm

1. Divide \(a\) by \(b\) to get quotient \(q\) and remainder \(r\).
2. If \(r = 0\), then \(b\) is considered the greatest common divisor (GCD).
3. If \(r neq 0\), replace \(a\) with \(b\) and \(b\) with \(r\), then repeat the process.

Example: Find GCD of 56 and 12

Step 1: Divide 56 by 12

\[ 56 = 12 times 4 + 8 \]

Step 2: Divide 12 by 8

\[ 12 = 8 times 1 + 4 \]

Step 3: Divide 8 by 4

\[ 8 = 4 times 2 + 0 \]

Since the remainder is 0, the GCD is 4.

Euclid's Division Algorithm is a fundamental technique to find the greatest common divisor efficiently. It plays a crucial role in number theory and is frequently applied in areas such as computer science, encryption techniques, and reducing fractions to their simplest form.