# GCD Formula

The Greatest Common Divisor (GCD) of two numbers is the largest possible number which divides both the numbers exactly.

The properties of GCD are as given below,

- GCD of two or more numbers divides each of the numbers without a remainder.
- GCD of two or more numbers is a factor of each of the numbers.
- GCD of two or more numbers is always less than or equal to each of the numbers.
- GCD of two or more prime numbers is 1 always.

## What is the GCD Formula?

There are 3 methods to calculate the GCD of two numbers:

- GCD by listing out the common factors
- GCD by prime factorization
- GCD by division method

Each of the above methods is explained in the given solved examples.

## Examples Using the GCD Formula

**Example 1:** What is the GCD of 30 and 42?

**Solution:**

List the factors of each number.

Factors of 30 - 1, 2, 3, 5, 6, 10, 15, 30

Factors of 42 - 1, 2, 3, 6, 7, 14, 21, 42

6 is the common factor and the greatest one.

Hence, the GCD of 30 and 42 is 6.

**Answer: **GCD of 30 and 42 is 6.

**Example 2:** What is the GCD of 60 and 90?

**Solution:**

Represent the numbers in the prime factored form.

60 = 2 × 2 × 3 × 5

90 = 2 × 3 × 3 × 5

GCD is the product of the factors that are common to each of the given numbers.

Thus, GCD of 60 and 90 = 2 × 3 × 5 = 30. ** **

**Answer: **GCD of 60 and 90 is 30.

**Example 3: **Find the GCD of 9000 and 980 using the "division method".

**Solution: **

Among the given numbers, 9000 is the smallest, and 980 is the largest.

We will divide the larger number by the smaller number.

Next, we will make the remainder as the divisor and the last divisor as the dividend and divide again.

We will repeat this process until the remainder is 0.

**Answer: **GCD of 9000 and 980 is 20.