Finding GCD by Prime Factorization

Last Updated : 23 Jul, 2025

Prime Factorization is the process of expressing a number as the product of its prime numbers. In other words, it involves breaking down a composite number into factors that are all prime numbers.

For example, the prime factorization of 60 is 2 × 2 × 3 × 5 or 2² × 3 × 5, since 2, 3, and 5 are primes. This method is widely used in finding the GCD, LCM, and simplifying fractions.

Steps to Find GCD by Prime Factorization

Here are the steps to find the GCD using prime factorization:

Step 1: Find the Prime Factorization of Each Number.

Example: For 24 and 36, the prime factorizations are 24 = 2³× 3 and 36 = 2²× 3².

Step 2: Identify the common prime factors and select the smallest power for each.

Example: Common factors are 2 and 3.

Step 3: Multiply: Multiply the lowest powers of each common factor to get the GCD.

Example: 2²× 3 = 4 × 3 = 12.

Thus, the GCD of 24 and 36 is 12.

Note: This method works for positive numbers only (natural numbers).

Here are some solved examples of GCD by prime factorization shown below:

Example 1: Find the GCD of 45 and 60.

Solution:

Prime Factorization:
45: 45 = 3²× 5
60: 60 = 2²× 3 × 5
Identify Common Factors:
The common factor is 3 and 5, with the lowest powers being 3¹ and 5¹.
Calculate GCD:
So, the GCD of 45 and 60 is:
GCD(45, 60) = 3¹× 5¹= 15.
Result: The GCD of 45, and 60 is 15.

Example 2: Find the GCD of 30, 60 and 84.

Solution:

Prime Factorization:
30: 30 = 2¹× 3¹× 5¹
60: 60 = 2²× 3¹× 5¹
84: 84 = 2²× 3¹× 7¹
Identify Common Factors:
The common factor is 2 and 3, with the lowest powers being 2¹ and 3¹.
Calculate GCD:
So, the GCD of 30, 60, and 84 is:
GCD(30, 60, 84) = 2¹× 3¹= 6.
Result: The GCD of 30, 60, and 84 is 6.