Problem 5 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ 24,14,21 $$

Step-by-Step Solution

Answer

The GCF of 24, 14, and 21 is 1.

Step 1: Understand the Problem

We need to find the greatest common factor (GCF) of the numbers 24, 14, and 21. The GCF is the largest integer that evenly divides each of the numbers without leaving a remainder.

Step 2: List the Factors of Each Number

First, we list all factors for each number. - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 14: 1, 2, 7, 14 - Factors of 21: 1, 3, 7, 21

Step 3: Identify Common Factors

Look for the common factors shared by all three numbers from the lists of factors found in the previous step. The common factors of 24, 14, and 21 are the numbers that appear in all three lists: 1 is the only number.

Step 4: Determine the GCF

The greatest common factor (GCF) is the largest number from the common factors identified. Since the only common factor is 1, the GCF of 24, 14, and 21 is 1.

Key Concepts

Factors of Numbers

To find the greatest common factor, we start by understanding what factors of numbers are. **Factors** are integers that can be multiplied together to produce the given number. For any given number, its factors divide it evenly without leaving any remainder.
For example:

• The factors of 24 include: 1, 2, 3, 4, 6, 8, 12, and 24. These numbers all divide 24 evenly.
• The factors of 14 are: 1, 2, 7, and 14.
• The factors of 21 are: 1, 3, 7, and 21.

Listing factors is the first step in finding the greatest common factor. Finding factors requires checking each number up to the given number itself. Try dividing the original number by smaller integers to check if it divides evenly.
Understanding the idea of factors helps us deconstruct numbers, making it easier to solve problems like finding the GCF.

Common Factors

Once we know the factors of each number, the next step is to find the **common factors**. A common factor is an integer factor that two or more numbers share. This means the same number divides each other number exactly without leaving a remainder.
In our example, when we found the factors of 24, 14, and 21, we looked at the lists like this:

• 24: 1, 2, 3, 4, 6, 8, 12, 24
• 14: 1, 2, 7, 14
• 21: 1, 3, 7, 21

From here, we identify which factors appear in every list. We see that the only number that appears in all three lists is 1, hence it is the only common factor in this case.
Finding common factors is crucial because it narrows down the choices to identify the largest possible factor shared by numbers, thus finding the GCF more efficiently.

Integer Division

**Integer division** is when one number is divided by another, and the result is an integer. This means there is no remainder left after the division. This concept is instrumental when determining factors and the greatest common factor (GCF).
For example, when calculating if 3 is a factor of 21, we perform the division 21 ÷ 3, which equals 7. Since the result is an integer with zero remainder, 3 is a factor of 21. Integer division also helps when checking for common factors across numbers. If you perform integer division across several numbers using the same divisor and each result yields no remainder, you can conclude that the divisor is a common factor.
Thus, the idea of doing division without remainders is key to understanding how different numbers relate and intersect through their factors. It illustrates how division not only determines multiplicity but also interconnectivity through shared factors.