GCD by Long Division Method (original) (raw)

Last Updated : 23 Jul, 2025

The Greatest Common Divisor (GCD), or Highest Common Factor (HCF), of two or more integers, is the largest positive integer that divides each of the integers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

In this article, we will explore using the Long Division Method, also known as Euclid's Division Algorithm, to find the Greatest Common Divisor (GCD).

Long Division Method (Euclid’s Division Algorithm) to Find the GCD (HCF)

This method uses Euclid's Division Algorithm that works for positive integers and follows these steps:

**Step 1: Use Euclid’s division lemma on two numbers a and b (where **a > b) to get the quotient **q and remainder **r such that a = bq + r and 0 ≤ r < b.

**Step 2: If r = 0, then b is the GCD of a and b. If r ≠ 0, apply the division lemma again using b and r as the new numbers.

**Step 3: Repeat this process until the remainder becomes zero.

**Step 4: Once the remainder is zero, the divisor at that step is the GCD of the original numbers.

This method is efficient, even for large numbers.

For example, we want to find the GCD of 12 and 56.

GCD by Long Division Method

Solved Examples on GCD/HCF by Long Division Method

Here are some solved examples of GCD by Long Division shown below:

**Example 1: Find the GCD of 45 and 60.

**Solution: To find the GCD of 45 and 60 using the long division method, we follow these steps:

**Divide 60 by 45:

• 60 ÷ 45 = 16 r**emainder 15.
• So, 60 = 45 × 1 + 15.

**Now divide 45 by the remainder 15:

• 45 ÷ 15 = 3 **remainder 0.
• So, 45 = 15 × 3.

Since the remainder is now 0, the divisor from the last division, **15, is the GCD.

**Therefore, the GCD of 45 and 60 is 15.

**Example 2: Find the GCD of 132, 198, and 264.

**Solution: To find the GCD of **132, 198, and 264 using the long division method, we follow these steps:

**Step 1: Find the GCD of 132 and 198

**Divide 198 by 132:

• 198 ÷ 132 = 1 **remainder 66.

So, 198 = 132 × 1 + 66.

**Now divide 132 by 66:

• 132 ÷ 66 = 2 **remainder 0.

Since the remainder is 0, the GCD of 132 and 198 is **66.

**Step 2: Find the GCD of 66 and 264

Now, we use 66 (the GCD of 132 and 198) and find the GCD with 264.

**Divide 264 by 66:

• 264 ÷ 66 = 4 **remainder 0.

Since the remainder is 0, the GCD of 66 and 264 is **66.

**Therefore, The GCD of 132, 198, and 264 is 66.