What is the Greatest Common Factor GCF Between 32x and 80xy?

The greatest common factor (GCF) between 32x and 80xy is 16x.

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. When dealing with algebraic expressions, the GCF extends to both the numerical coefficients and the variable parts. It is the product of the GCF of the coefficients and the GCF of the variables.

Step-by-Step Calculation of GCF(32x, 80xy)

To determine the GCF of algebraic expressions like 32x and 80xy, we break down the process into two key parts: finding the GCF of the numerical coefficients and then finding the GCF of the variable components.

1. Finding the GCF of the Numerical Coefficients (32 and 80)

The GCF of two non-zero integers, such as 32 and 80, is the greatest positive integer that divides both numbers completely. For these specific numbers, that value is 16. This means 16 is the largest number that 32 and 80 can both be divided by without any remainder.

To illustrate, consider the factors of each number:

Number	Factors
32	1, 2, 4, 8, 16, 32
80	1, 2, 4, 5, 8, 10, 16, 20, 40, 80

By examining the factors, we can clearly see that the largest number common to both lists is 16.

2. Finding the GCF of the Variable Parts (x and xy)

To find the GCF of the variable parts in algebraic terms, identify all variables that are common to every term. For each common variable, use its lowest power (exponent) found across all terms.

In the term 32x, the variable part is x (which is x¹).
In the term 80xy, the variable part is xy (which is x¹y¹).

Here's how we determine the GCF of the variables:

The variable x is present in both 32x and 80xy. The lowest power of x is x¹.
The variable y is only present in 80xy and not in 32x. Therefore, y is not a common variable and is not included in the GCF of the variable parts.

Thus, the GCF of the variable parts x and xy is x.

3. Combining the GCFs

To arrive at the complete GCF of the algebraic expressions, multiply the GCF of the numerical coefficients by the GCF of the variable parts.

GCF(32x, 80xy) = GCF(32, 80) × GCF(x, xy)
GCF(32x, 80xy) = 16 × x
GCF(32x, 80xy) = 16x

Why is GCF Important?

Understanding and calculating the GCF is a fundamental skill in algebra and has several practical applications:

Simplifying Algebraic Expressions: GCF is essential for factoring expressions. For instance, the expression 32x + 80xy can be factored by its GCF, 16x, to become 16x(2 + 5y), making the expression simpler and often easier to analyze or solve.
Solving Equations: Factoring expressions using the GCF can be a crucial step in solving polynomial equations.
Simplifying Fractions: Just as with numerical fractions, GCF is used to reduce algebraic fractions to their simplest form.