Greatest Common Divisor Formula
Before going to learn the greatest common divisor formula, let us recall what is the Greatest Common Divisor (GCD). The GCD of two numbers is the largest possible number that 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 Greatest Common Divisor Formula?
We do not have any greatest common divisor formula. Instead, there are 3 methods to calculate the GCF of two numbers:

Greatest Common Divisor by listing out the common factors

Greatest Common Divisor by prime factorization

Greatest Common Divisor by division method
Each of the above methods is explained in the given solved examples.
Solved Examples Using Greatest Common Divisor Formula

Example 1: What is the Greatest Common Divisor 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 greatest common divisor of 30 and 42 is 6.
Answer: Greatest common divisor of 30 and 42 is 6

Example 2: What is the Greatest Common Divisor of 60 and 90?
Solution:
Represent the numbers in the prime factored form.
\[\begin{align} 60&= 2 \times 2 \times 3 \times 5\\ 90&=2 \times 3\times 3 \times 5 \end{align} \]
Greatest common divisor is the product of the factors that are common to each of the given numbers.
Thus, greatest common divisor of 60 and 90 = 2 x 3 x 5 = 30.
Answer: Greatest common divisor of 60 and 90 is 30

Example 3: Find the Greatest Common Divisor 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: Greatest common divisor of 9000 and 980 is 20.