Highest Common Factor Of 48 And 36

Finding the Highest Common Factor (HCF) of 48 and 36: A Step-by-Step Guide

Finding the highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving algebraic problems. This article will guide you through several methods to determine the HCF of 48 and 36, explaining the process clearly and concisely. Understanding these methods will equip you to find the HCF of any pair of numbers.

What is the Highest Common Factor (HCF)? The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In simpler terms, it's the biggest number that's a factor of both numbers.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. The prime factors are the prime numbers that multiply together to make the original number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself.

1. Prime Factorization of 48: 48 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

2. Prime Factorization of 36: 36 = 2 x 2 x 3 x 3 = 2² x 3²

3. Identifying Common Factors: Look for the common prime factors in both factorizations. Both 48 and 36 have two factors of 2 and one factor of 3.

4. Calculating the HCF: Multiply the common prime factors together: 2 x 2 x 3 = 12

Therefore, the highest common factor of 48 and 36 is 12.

Method 2: Listing Factors

This method is suitable for smaller numbers. List all the factors of each number and then identify the largest factor common to both.

1. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

2. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

3. Common Factors: The common factors of 48 and 36 are 1, 2, 3, 4, 6, and 12.

4. Highest Common Factor: The largest common factor is 12.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a more efficient method for larger numbers. It uses repeated division until the remainder is 0. The last non-zero remainder is the HCF.

1. Divide the larger number (48) by the smaller number (36): 48 ÷ 36 = 1 with a remainder of 12.
2. Replace the larger number with the smaller number (36) and the smaller number with the remainder (12): 36 ÷ 12 = 3 with a remainder of 0.
3. Since the remainder is 0, the HCF is the last non-zero remainder, which is 12.

Conclusion:

Regardless of the method used, the highest common factor of 48 and 36 is 12. Choosing the best method depends on the size of the numbers and your comfort level with different mathematical techniques. Understanding the concept of HCF and mastering these methods is crucial for various mathematical applications and problem-solving scenarios. Now you can confidently tackle similar problems involving finding the HCF of other number pairs!