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Chapter 5: Problem 4

Find the greatest common factor for each list of numbers. 18,24

Short Answer

Expert verified
The GCF is 6.

Step by step solution

01

List the factors

First, list all the factors of each number. Factors of 18: 1, 2, 3, 6, 9, 18Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
02

Identify the common factors

Next, identify the factors that appear in both lists. Common factors: 1, 2, 3, 6
03

Determine the greatest common factor (GCF)

From the common factors, find the largest number. Greatest Common Factor (GCF): 6

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Key Concepts

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

factors
Factors are numbers that divide another number completely without leaving a remainder. For instance, we can determine that 2 is a factor of 18 because when we divide 18 by 2, the result is 9, and nothing is left over.

To find the factors of a number:
  • Start with 1 and the number itself.
  • Check each number up to half of the original number to see if it divides evenly.
  • List all these numbers as they are the factors.
For example:
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Understanding factors is the first step in finding the Greatest Common Factor (GCF).
common factors
Common factors are numbers that are factors of two or more numbers. Identifying common factors helps us understand the relationship between different numbers.

To find common factors:
  • List all factors of each number.
  • Look for factors that appear in all lists.
  • These shared numbers are your common factors.
For the numbers 18 and 24:
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Common factors: 1, 2, 3, 6
Recognizing common factors is essential for the next step, finding the Greatest Common Factor (GCF).
GCF calculation
The Greatest Common Factor (GCF), sometimes called the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder.

Finding the GCF involves:
  • Listing all factors of each number.
  • Identifying the common factors.
  • Selecting the largest number from these common factors.
For example, with the numbers 18 and 24:
  • Common factors: 1, 2, 3, 6
  • Greatest common factor: 6
Knowing how to calculate the GCF can make simplifying fractions and solving division problems easier.

In more complex cases, you can use prime factorization to find the GCF. This method involves breaking each number into its prime factors and then multiplying the common prime factors together.

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Most popular questions from this chapter

Factor each binomial completely. \(125 t^{3}+8 s^{3}\)

Find all integers \(k\) so that the trinomial can be factored using the methods of this section. \(2 x^{2}+k x-3\)

Factor each trinomial. \(-x^{2}+x+72\)

Solve each problem. Find two consecutive odd integers such that their product is 15 more than three times their sum.

Factor each trinomial completely. \(6 m^{6} n+7 m^{5} n^{2}+2 m^{4} n^{3}\)
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