Greatest Common Factor Of 48 And 54

Finding the Greatest Common Factor (GCF) of 48 and 54

This article will guide you through finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of 48 and 54. Understanding how to calculate the GCF is a fundamental skill in mathematics, useful in various applications from simplifying fractions to solving algebraic equations. We'll explore two common methods: prime factorization and the Euclidean algorithm. By the end, you'll be able to confidently determine the GCF of any two numbers.

What is the Greatest Common Factor (GCF)?

The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into all of them without leaving a remainder. It's the largest shared factor among the numbers. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The GCF of 12 and 18 is 6 because it's the largest number that divides both 12 and 18 without leaving a remainder.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

1. Find the prime factorization of 48:

   48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

2. Find the prime factorization of 54:

   54 = 2 x 27 = 2 x 3 x 9 = 2 x 3 x 3 x 3 = 2 x 3³

3. Identify common prime factors:

   Both 48 and 54 share one factor of 2 and three factors of 3.

4. Calculate the GCF:

   The GCF is the product of the lowest powers of the common prime factors. In this case, it's 2¹ x 3¹ = 2 x 3 = 6.

Therefore, the greatest common factor of 48 and 54 is 6.

Method 2: Euclidean Algorithm

The Euclidean algorithm provides a more efficient method for larger numbers. It's based on repeated division.

1. Divide the larger number (54) by the smaller number (48):

   54 ÷ 48 = 1 with a remainder of 6.

2. Replace the larger number with the smaller number (48) and the smaller number with the remainder (6):

   Now we divide 48 by 6.

3. Repeat the process:

   48 ÷ 6 = 8 with a remainder of 0.

4. The GCF is the last non-zero remainder:

   The last non-zero remainder is 6.

Therefore, using the Euclidean algorithm, the greatest common factor of 48 and 54 is also 6.

Conclusion:

Both methods, prime factorization and the Euclidean algorithm, successfully determine the greatest common factor of 48 and 54 as 6. Choose the method that best suits your comfort level and the complexity of the numbers involved. Understanding the GCF is a valuable skill in various mathematical contexts and problem-solving scenarios.