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Chapter 6: Problem 6

Find the greatest common factor for each list of numbers. \(20,22,23\)

Short Answer

Expert verified
The greatest common factor is 1.

Step by step solution

01

Title - List All Factors

List all the factors of each number. The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 22 are 1, 2, 11, and 22. The factors of 23 are 1 and 23.
02

Title - Identify Common Factors

Identify the common factors among the lists. The common factors among 20, 22, and 23 are only 1.
03

Title - Greatest Common Factor

The greatest number in the common factors list is the greatest common factor. Hence, the greatest common factor of 20, 22, and 23 is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factors
To understand factors, you need to grasp that a factor of a number divides that number exactly, without leaving a remainder. For example, the factors of 20 are 1, 2, 4, 5, 10, and 20.

This means each of these numbers divides 20 evenly. It's important to list all factors for each number in your problem first because this step helps you identify common factors later.

When you are working with factors:
  • Always start with 1 and the number itself as factors.
  • Try dividing the number by smaller numbers to see if they divide evenly.
For instance, with 20, you see that 20 can be divided by 2 (as it鈥檚 an even number) giving us factors like 2 and 10. Keep going systematically until you list all factors.
Common Factors
Common factors can be found once you have listed all factors of each number in your set. They are simply the numbers that appear in all lists. From our example, we had the numbers:
  • 20 with factors: 1, 2, 4, 5, 10, 20
  • 22 with factors: 1, 2, 11, 22
  • 23 with factors: 1, 23
By comparing these lists, you see the only common factor is 1. Finding common factors involves:
  • Comparing the factor lists side by side
  • Identifying which numbers appear in all lists
This step might seem repetitive, but it鈥檚 essential to ensure you correctly identify the greatest common factor later.
Greatest Common Factor
The greatest common factor (GCF) is the largest number that appears in all sets of factors. From our above steps, we determined that 1 is the only common factor between 20, 22, and 23.
Therefore, the GCF is 1 because it is the greatest/common factor listed. To find the GCF, follow these tips:
  • List all the factors of each number accurately.
  • Identify the common numbers in all lists.
  • Select the highest number from the common factors.
Remember the process: list, compare, and select. This method works for any set of numbers, helping you to consistently find the GCF.

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