Highest Common Factor Of 72 And 96

Finding the Highest Common Factor (HCF) of 72 and 96: 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 72 and 96, explaining each step clearly and concisely. Understanding these methods will equip you to find the HCF of any two numbers efficiently.

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. For example, the HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.

Methods to Find the HCF of 72 and 96

We'll explore three common methods: prime factorization, listing factors, and the Euclidean algorithm.

1. Prime Factorization Method

This method involves breaking down each number into its prime factors. The HCF is then found by multiplying the common prime factors raised to the lowest power.

• Prime factorization of 72: 2 x 2 x 2 x 3 x 3 = 2³ x 3²
• Prime factorization of 96: 2 x 2 x 2 x 2 x 2 x 3 = 2⁵ x 3¹

Both numbers share three 2s and one 3. Therefore, the HCF is 2³ x 3¹ = 8 x 3 = 24.

2. Listing Factors Method

This method involves listing all the factors of each number and identifying the largest common factor.

• Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

By comparing the lists, we find that the largest common factor is 24. This method is effective for smaller numbers but becomes cumbersome for larger ones.

3. Euclidean Algorithm Method

This is a more efficient method for larger numbers. It involves repeatedly applying the division algorithm until the remainder is zero. The last non-zero remainder is the HCF.

1. Divide the larger number (96) by the smaller number (72): 96 ÷ 72 = 1 with a remainder of 24.
2. Replace the larger number with the smaller number (72) and the smaller number with the remainder (24).
3. Repeat: 72 ÷ 24 = 3 with a remainder of 0.
4. Since the remainder is 0, the HCF is the last non-zero remainder, which is 24.

The Euclidean algorithm provides a systematic and efficient way to find the HCF, especially for larger numbers where listing factors becomes impractical.

Conclusion:

We've explored three different methods to find the highest common factor of 72 and 96, all yielding the same result: 24. Choosing the best method depends on the numbers involved and your preference. The Euclidean algorithm is generally preferred for larger numbers due to its efficiency. Understanding these methods will be beneficial for solving various mathematical problems involving common factors and divisors. Remember to practice these methods to improve your proficiency and understanding of this fundamental mathematical concept.