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Chapter 5: Problem 9

Find the greatest common factor for each list of numbers. 4,9,12

Short Answer

Expert verified
The greatest common factor is 1.

Step by step solution

01

- Definition

The greatest common factor (GCF) is the largest number that divides each of the given numbers without leaving a remainder.
02

- List the Factors of Each Number

List all the factors of each number:- Factors of 4: 1, 2, 4- Factors of 9: 1, 3, 9- Factors of 12: 1, 2, 3, 4, 6, 12
03

- Identify Common Factors

Find the factors that are common to all the lists:- Common Factors: 1
04

- Find the Greatest Common Factor

From the common factors, identify the largest one:- Greatest Common Factor: 1

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Key Concepts

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

Factors
To understand the concept of finding the greatest common factor (GCF), it's essential to first learn about factors. Factors are numbers you can multiply together to get another number. For instance, the factors of 4 are 1, 2, and 4, because 1 x 4 = 4 and 2 x 2 = 4.
To find the factors of a number, start by dividing the number by 1, and then continue with all the integers up to that number. If the division results in a whole number, then both the divisor and the quotient are factors of the original number.
For example:
  • Factors of 4: 1, 2, 4
  • Factors of 9: 1, 3, 9
  • Factors of 12: 1, 2, 3, 4, 6, 12
Common Factors
Common factors are the numbers that are factors of two or more numbers. To find the common factors, you need to list out all the factors for each number and then identify the numbers that appear in all the lists.
For example, let's find the common factors of 4, 9, and 12:
  • Factors of 4: 1, 2, 4
  • Factors of 9: 1, 3, 9
  • Factors of 12: 1, 2, 3, 4, 6, 12
You can see that the only factor common to all three numbers is 1. Therefore, the common factor is 1.
Divisibility
Divisibility is a helpful concept when determining factors and, subsequently, the greatest common factor. A number is divisible by another number if, when you divide them, the result is a whole number without any remainder. For example, 12 is divisible by 3 because when you divide 12 by 3, you get 4, which is a whole number.
Understanding divisibility rules can make finding factors easier:
  • A number is divisible by 2 if it is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 9 if the sum of its digits is divisible by 9.
Applying these rules will help you quickly find the factors of any number, which then makes it easier to find common factors and, ultimately, the greatest common factor.

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Most popular questions from this chapter

Factor each binomial completely. \(27 w^{3}-216 z^{3}\)

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