Finding the Highest Common Factor (H.C.F.)

The Highest Common Factor (H.C.F.), also known as the Greatest Common Divisor (GCD), is the largest number or expression that divides two or more numbers or expressions without leaving a remainder. Let’s explore how to find the H.C.F. of $2y^2$ and $y$

Understanding Factors

Before we dive into finding the H.C.F., let’s understand the concept of factors. Factors are numbers or expressions that divide another number or expression evenly.

Example: The factors of 12 are 1, 2, 3, 4, 6, and 12 because they all divide 12 evenly.

Finding the H.C.F. of $2y^2$ and $y$

1. Factorize each expression:

   • $2y^2$ can be factored as $2 times y times y$
   • $y$ can be factored as $1 times y$

2. Identify common factors:

   Both expressions have the factor $y$ in common.

3. Multiply the common factors:

   The H.C.F. is the product of the common factors. In this case, the H.C.F. is simply $y$

Explanation

Let’s break down why $y$ is the H.C.F. of $2y^2$ and $y$:

• Divisibility: $y$ divides $2y^2$ evenly, resulting in $2y$. It also divides $y$ evenly, resulting in 1.
• Largest Factor: There’s no larger expression that can divide both $2y^2$ and $y$ without leaving a remainder.

Real-World Example

Imagine you have two pieces of fabric: one is $2y^2$ inches long and the other is $y$ inches long. You want to cut both pieces into smaller pieces of equal length, with the largest possible length for each piece. The H.C.F., $y$, represents the largest length you can cut each piece into, ensuring no fabric is wasted.

Summary

The H.C.F. of $2y^2$ and $y$ is $y$. This means that $y$ is the largest expression that divides both $2y^2$ and $y$ evenly. Understanding the concept of H.C.F. helps us solve various problems in algebra and other areas of mathematics.

3. BBC Bitesize – Finding the Highest Common Factor

Citations

1. 1. Math is Fun – Highest Common Factor
2. 2. Khan Academy – Greatest Common Factor