what is the greatest common factor of 8x and 40y

2 min read 10-03-2025

what is the greatest common factor of 8x and 40y

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides evenly into two or more numbers. Let's find the GCF of 8x and 40y.

Understanding Factors

Before we find the GCF, let's refresh what factors are. Factors are numbers that divide evenly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Finding the GCF of 8x and 40y

To find the GCF of 8x and 40y, we'll look at the factors of both terms separately, then identify the largest common factor.

Step 1: Find the factors of 8x

The factors of 8 are 1, 2, 4, and 8. Since 8x includes the variable 'x', we also consider 'x' as a factor. Therefore, the factors of 8x are 1, 2, 4, 8, x, 2x, 4x, and 8x.

Step 2: Find the factors of 40y

The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. Since 40y includes the variable 'y', we also consider 'y' as a factor. Therefore, the factors of 40y are 1, 2, 4, 5, 8, 10, 20, 40, y, 2y, 4y, 5y, 8y, 10y, 20y, and 40y.

Step 3: Identify the common factors

Now, let's compare the factors of 8x and 40y and identify the common factors:

1: Both 8x and 40y are divisible by 1.
2: Both 8x and 40y are divisible by 2.
4: Both 8x and 40y are divisible by 4.
8: Both 8x and 40y are divisible by 8.

Step 4: Determine the greatest common factor

From the common factors, we see that the largest number that divides both 8x and 40y evenly is 8. Therefore, the greatest common factor of 8x and 40y is $\boxed{8}$ .

Applying the GCF: Simplifying Expressions

Understanding the GCF is crucial in simplifying algebraic expressions. For example, if you have the expression 8x + 40y, you can factor out the GCF (8) to simplify it:

8x + 40y = 8(x + 5y)

This simplified form is often easier to work with in further mathematical operations.

Practice Problems

Try finding the GCF for these pairs of terms:

12a and 18b
25m and 15n
36p and 48q

Remember to consider both numerical and variable factors when finding the GCF. The GCF is always the largest number and variable combination that divides both terms evenly.