Highest Common Factor Of 36 And 84

Finding the Highest Common Factor (HCF) of 36 and 84: 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. This article will guide you through different methods to determine the HCF of 36 and 84, explaining the process clearly and concisely. Understanding HCF is crucial for simplifying fractions, solving algebraic equations, and various other mathematical applications. This guide will cover prime factorization and the Euclidean algorithm, providing you with a comprehensive understanding of this important mathematical function.

What is the Highest Common Factor (HCF)?

The highest common factor (HCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It represents the greatest common divisor shared by those numbers. For example, finding the HCF of 36 and 84 will tell us the largest number that perfectly divides both 36 and 84.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

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

2. Prime Factorization of 84: 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7

3. Identifying Common Factors: Compare the prime factorizations of both numbers. We see that both 36 and 84 share 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 HCF of 36 and 84 is 12.

Method 2: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the HCF, particularly useful for larger numbers. It uses repeated division with remainder.

1. Divide the larger number by the smaller number: 84 ÷ 36 = 2 with a remainder of 12.

2. Replace the larger number with the smaller number, and the smaller number with the remainder: Now we find the HCF of 36 and 12.

3. Repeat the division: 36 ÷ 12 = 3 with a remainder of 0.

4. The HCF is the last non-zero remainder: Since the remainder is 0, the HCF is the previous remainder, which is 12.

Conclusion:

Both the prime factorization method and the Euclidean algorithm effectively determine the highest common factor. The prime factorization method provides a clear visual representation of the shared factors, while the Euclidean algorithm offers a more efficient approach for larger numbers. In both cases, we find that the highest common factor of 36 and 84 is 12. This means 12 is the largest number that perfectly divides both 36 and 84. Understanding these methods will equip you with the skills to solve similar problems involving finding the greatest common divisor of any two integers.