Book cover for Algebra A Combined Function

Algebra A Combined Function

Elayn Martin-Gay

ISBN #9780321726391

4th Edition

6,670 Questions

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28,872 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces essential concepts in factoring numbers and expressions. Students learn to list all factors of natural numbers, distinguish between prime and composite numbers, and perform prime factorization. Understanding these concepts is crucial for calculating the least common multiple (LCM) and supports further algebraic manipulation and problem-solving.

Learning Objectives

1

Identify and list the factors of natural numbers.

2

Distinguish between prime numbers and composite numbers.

3

Perform prime factorization of a given number.

4

Calculate the least common multiple (LCM) using factors and prime factorizations.

Key Concepts

CONCEPT

DEFINITION

Factor

A number that divides another number evenly without leaving a remainder.

Prime Number

A natural number greater than 1 with exactly two factors: 1 and itself.

Composite Number

A natural number greater than 1 that has more than two factors.

Natural Numbers

Counting numbers starting from 1, 2, 3, and so on.

Prime Factorization

Expressing a number as a product of its prime factors. This factorization is unique regardless of the order of factors.

Least Common Multiple (LCM)

The smallest multiple that is exactly divisible by each number in a set.

Example Problems

Example 1

List the factors of each number. See Examples 1 and $2 .$ 9

Example 2

List the factors of each number. See Examples 1 and $2 .$ 8

Example 3

List the factors of each number. See Examples 1 and $2 .$ 24

Example 4

List the factors of each number. See Examples 1 and $2 .$ 36

Example 5

List the factors of each number. See Examples 1 and $2 .$ 42
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Step-by-Step Explanations

QUESTION

How do you list all factors of 20?

STEP-BY-STEP ANSWER:

Step 1: Identify pairs of numbers that multiply to 20. Start with 1 and 20 because 1 x 20 = 20.
Step 2: Check the next integer, 2. Since 20 ÷ 2 = 10 (an integer), 2 and 10 are factors.
Step 3: Check 3. Because 20 ÷ 3 is not an integer, 3 is not a factor.
Step 4: Check 4. Since 20 ÷ 4 = 5, include 4 and 5 as factors.
Step 5: Once the square root of 20 is exceeded, you have found all factor pairs.
Final Answer: The factors of 20 are 1, 2, 4, 5, 10, and 20.

Listing Factors of a Number (e.g., 20)

QUESTION

What are the steps to find the prime factorization of 80?

STEP-BY-STEP ANSWER:

Step 1: Express 80 as a product of two numbers. For example, 80 = 8 x 10.
Step 2: Factor 8 into primes: 8 = 2 x 2 x 2.
Step 3: Factor 10 into primes: 10 = 2 x 5.
Step 4: Combine the factors: 80 = 2 x 2 x 2 x 2 x 5.
Final Answer: The prime factorization of 80 is 2 x 2 x 2 x 2 x 5.

Prime Factorization (e.g., 80)

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Common Mistakes

  • Missing factor pairs when listing factors, leading to an incomplete list.
  • Confusing prime numbers with composite numbers by overlooking additional factors.
  • Not fully breaking down composite factors into primes in the process of prime factorization.
  • Believing that the order of multiplication in prime factorization affects its uniqueness.