Problem 22 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of numbers. $$ 16,24,48 $$

Step-by-Step Solution

Answer

The GCF of 16, 24, and 48 is 8.

Step 1: List the Factors of Each Number

First, identify all the factors of each number in the list. - Factors of 16: 1, 2, 4, 8, 16 - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Identify the Common Factors

Next, find the common factors among all three numbers from the lists you created. - Common factors of 16, 24, and 48: 1, 2, 4, 8

Step 3: Determine the Greatest Common Factor

From the list of common factors, choose the largest one. This is called the Greatest Common Factor (GCF). - Greatest Common Factor: 8

Key Concepts

Factors

Factors are the numbers you can multiply together to get another number. They are crucial in breaking down numbers into smaller, manageable parts.

Each number has a set of factors, which are useful when you're trying to find commonality or simplify problems. For example, when looking for factors of a number:

• Start from 1 and divide the number by other integers to see if it divides evenly.
• For 16, these would be 1, 2, 4, 8, and 16, since each divides 16 without a remainder.

Understanding factors forms the foundation of finding the Greatest Common Factor (GCF). Knowing the complete list of factors for each involved number enhances clarity and precision in mathematical problem-solving.

Common Factors

Common factors are factors that two or more numbers have in common. They are helpful to find when solving problems that involve multiple numbers, such as finding the GCF.

Once you have identified the factors of your numbers:

• Check each list of factors and highlight numbers that appear in each list.
• For instance, the numbers 1, 2, 4, and 8 are common across the factors of 16, 24, and 48.

Spotting common factors allows us to understand and compare different numbers at a fundamental level, preparing us for finding the largest one, the GCF.

Step-by-Step Solution

Solving problems involving the Greatest Common Factor involves a systematic approach. Breaking down the problem step-by-step makes it handleable.

First, by listing the factors, you get a clear map of each number's possible divisors. This sets the stage for selecting common factors. When you find that certain numbers appear across all lists, you've uncovered your common factors.

Finally, determining the Greatest Common Factor is a matter of selecting the highest number from your common factors. In our example, this is 8.

• This process not only finds the GCF but also reinforces understanding of numerical relationships.
• It sharpens the ability to identify patterns and draw conclusions based on them.

Taking a step-by-step approach ensures thoroughness and accuracy in solving mathematical problems.