# 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:

- 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.

## Solved 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.

\[\begin{align} 60&= 2 \times 2 \times 3 \times 5\\ 90&=2 \times 3\times 3 \times 5 \end{align} \]

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

Thus, GCD of 60 and 90 = 2 x 3 x 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.