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 $y$
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Factorize each expression:
- $2y^2$ can be factored as $2 times y times y$
- $y$ can be factored as $1 times y$
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Identify common factors:
Both expressions have the factor $y$ in common.
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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