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Chapter 10: Problem 11

Determine the greatest common factor. 12 and 18

Short Answer

Expert verified
The greatest common factor of 12 and 18 is 6.

Step by step solution

01

List the Factors of Each Number

First, list all the factors of 12. Factors of 12 are: 1, 2, 3, 4, 6, 12.Next, list all the factors of 18. Factors of 18 are: 1, 2, 3, 6, 9, 18.
02

Identify Common Factors

Compare the lists of factors of 12 and 18 to see which numbers are common to both lists. Common factors of 12 and 18 are: 1, 2, 3, 6.
03

Determine the Greatest Common Factor

Among the common factors identified in Step 2 (1, 2, 3, 6), the greatest number is 6. Therefore, the greatest common factor (GCF) of 12 and 18 is 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 exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12. Each of these numbers can divide 12 perfectly.
To find factors:
  • Start from 1 and try dividing the number by 1, 2, 3, and so on.
  • If the division results in a whole number, then both the divisor and the quotient are factors.
For instance, dividing 12 by 3 results in 4, meaning both 3 and 4 are factors of 12.
Common Factors
Common factors are the numbers that are factors of two or more numbers. For instance, when we look at the factors of 12 (1, 2, 3, 4, 6, 12) and the factors of 18 (1, 2, 3, 6, 9, 18), the common factors are numbers present in both lists. Here, the common factors of 12 and 18 are 1, 2, 3, and 6.
To find common factors:
  • List out all the factors of both numbers.
  • Identify the numbers that appear in both lists.
In this exercise, the result is the list {1, 2, 3, 6}.
Greatest Common Divisor
The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest number that is a common factor of two or more numbers. From the previous section, we identified the common factors of 12 and 18 as 1, 2, 3, and 6.
The GCD is the highest number in this set of common factors. In this case, 6 is the largest number, hence the GCD of 12 and 18 is 6.
Steps to determine the GCD:
  • List all factors of the given numbers.
  • Identify the common factors.
  • Select the largest number from the common factors.
Prealgebra
Prealgebra sets the foundation for algebra, introducing basic arithmetic operations and properties of numbers. Key concepts include understanding factors, common factors, and the greatest common divisor. These basic skills are crucial as they help in simplifying fractions, finding least common multiples, and solving problems involving ratios.
In the example exercise, identifying the GCF helps reinforce the understanding of these fundamental concepts. It also prepares students for more advanced problem-solving in algebra.

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