Problem 63 - Algebra 2

Question

Find the greatest common factor of each set of numbers. $$ 16,28 $$

Step-by-Step Solution

Answer

The greatest common factor of 16 and 28 is 4.

Step 1: Understanding the Problem

We need to find the greatest common factor (GCF) of the numbers 16 and 28. The GCF is the largest number that divides into each of the numbers without leaving a remainder.

Step 2: List the Factors of Each Number

First, list all the factors of 16 and 28. - Factors of 16 are: 1, 2, 4, 8, 16. - Factors of 28 are: 1, 2, 4, 7, 14, 28.

Step 3: Identify Common Factors

Now, identify the factors that 16 and 28 have in common. These are the numbers that appear on both lists: 1, 2, 4.

Step 4: Determine the Greatest Common Factor

From the list of common factors (1, 2, 4), the greatest factor is 4. Therefore, the greatest common factor of 16 and 28 is 4.

Key Concepts

Factors

In mathematics, factors of a number are the integers or whole numbers that can be multiplied together to produce the original number. Finding factors is like reverse multiplication. For instance, if you're looking at the number 16, you'd say:

• 1 and 16 are factors because 1 multiplied by 16 equals 16.
• 2 and 8 are factors because 2 multiplied by 8 equals 16.
• 4 is also a factor, because 4 multiplied by 4 equals 16.

If a number "fits" into another number without leaving any remainder, then it's a factor of that number. Every number has 1 and itself as factors automatically. Exploring factors is a key first step to understanding more complex math ideas like greatest common factors, which build on this basic concept.

Common Factors

Common factors are factors that two or more numbers share. If you look at the factors of both numbers and pick out the ones they have in common, you've got your set of common factors. Let's break it down using 16 and 28 as examples again:

• For 16, the factors are 1, 2, 4, 8, and 16.
• For 28, the factors are 1, 2, 4, 7, 14, and 28.

To find the common factors of 16 and 28, look at both lists and identify the numbers that appear in both. You see 1, 2, and 4 appear in each list—they are the common factors. By comparing both sets of factors, you notice that certain numbers, such as 8 and 7, don't match up, so they're not common. Focusing on common factors is important because they tell us about the numbers’ shared properties, which is quite useful in reducing fractions or solving other math problems.

Divisibility

Divisibility refers to the ability of one number to be divided by another without leaving a remainder. In simpler terms, when a number can be "evenly divided" by another, it is divisible by it. Here's how it works:

• If 8 divided by 2 equals 4 with no remainder, 8 is divisible by 2.
• If you try 9 divided by 2 and get 4 with a remainder of 1, then 9 is not completely divisible by 2.

When finding the greatest common factor, checking divisibility helps confirm that you have indeed found a factor. For the numbers 16 and 28, you tested divisibility to ensure that numbers like 2 and 4 truly divide both numbers without leftover pieces. Understanding divisibility is essential for identifying factors and solving various math puzzles like finding common multiples and simplifying numbers.