Highest Common Factor Of 15 And 18

Finding the Highest Common Factor (HCF) of 15 and 18: 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 the process of determining the HCF of 15 and 18 using different methods, making it easy to understand for both beginners and those looking to refresh their knowledge. Understanding HCF is crucial for various mathematical operations and problem-solving scenarios.

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. It's the greatest common divisor that all the numbers share. In simpler terms, it's the biggest number that goes into both numbers evenly.

Methods to Find the HCF of 15 and 18

We can use several methods to find the HCF of 15 and 18:

1. Listing Factors Method

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

• Factors of 15: 1, 3, 5, 15
• Factors of 18: 1, 2, 3, 6, 9, 18

Comparing the lists, we see that the common factors are 1 and 3. The largest of these common factors is 3.

Therefore, the HCF of 15 and 18 is 3.

2. Prime Factorization Method

This method involves expressing each number as a product of its prime factors. The HCF is then found by multiplying the common prime factors raised to their lowest powers.

• Prime factorization of 15: 3 x 5
• Prime factorization of 18: 2 x 3 x 3 = 2 x 3²

The only common prime factor is 3, and its lowest power is 3¹. Therefore, the HCF is 3.

3. Euclidean Algorithm Method

The Euclidean algorithm is a highly efficient method for finding the HCF of two numbers, especially for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the HCF.

1. Divide the larger number (18) by the smaller number (15): 18 ÷ 15 = 1 with a remainder of 3.
2. Replace the larger number with the smaller number (15) and the smaller number with the remainder (3): 15 ÷ 3 = 5 with a remainder of 0.
3. Since the remainder is 0, the HCF is the last non-zero remainder, which is 3.

Conclusion:

Using any of these methods—listing factors, prime factorization, or the Euclidean algorithm—we find that the highest common factor of 15 and 18 is 3. The choice of method depends on the complexity of the numbers involved and personal preference. The Euclidean algorithm is generally preferred for larger numbers due to its efficiency. Understanding how to calculate the HCF is a valuable skill in various mathematical applications.