Learn on PengiBig Ideas Math, Course 1Chapter 1: Numerical Expressions and Factors

Lesson 4: Prime Factorization

In this Grade 6 lesson from Big Ideas Math, Course 1, Chapter 1, students learn how to find the prime factorization of composite numbers by applying divisibility rules for 2, 3, 5, 6, 9, and 10 and using factor trees. Students practice breaking numbers down into their prime factors and expressing results using exponential notation, such as writing 60 as 2² × 3 × 5. The lesson builds foundational skills for working with common factors and multiples as outlined in Common Core Standard 6.NS.4.

Section 1

Systematically Find All Factor Pairs

Property

To find all whole number factor pairs for an area AA, systematically test integers n=1,2,3,…n=1, 2, 3, \dots to see if they divide AA evenly.
If A÷n=mA \div n = m with no remainder, then (n,m)(n, m) is a factor pair.
The process can be stopped when nn becomes greater than mm.

Examples

Section 2

Procedure: Find Prime Factorization with a Factor Tree

Property

To find the prime factorization of a composite number using the tree method:
Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
Step 2. If a factor is prime, that branch is complete. Circle the prime.
Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
Step 4. Write the composite number as the product of all the circled primes.

Examples

  • To factor 36, start with 6⋅66 \cdot 6. Break the first 6 into 2⋅32 \cdot 3 and circle both primes. Break the second 6 into 2⋅32 \cdot 3 and circle both primes. The factorization is 22⋅322^2 \cdot 3^2, or 2⋅2⋅3⋅32 \cdot 2 \cdot 3 \cdot 3.
  • For 28, branch into 4⋅74 \cdot 7. Circle the prime 7. Branch 4 into 2⋅22 \cdot 2 and circle both 2s. The result is 22⋅72^2 \cdot 7, or 2⋅2⋅72 \cdot 2 \cdot 7.
  • To factor 60, you can start with 6⋅106 \cdot 10. Branch 6 into 2⋅32 \cdot 3 and 10 into 2⋅52 \cdot 5. All are prime, so the factorization is 22⋅3⋅52^2 \cdot 3 \cdot 5, or 2⋅2⋅3⋅52 \cdot 2 \cdot 3 \cdot 5.

Explanation

This visual method lets you branch out from a number, splitting it into pairs of factors. You keep branching until every path ends in a prime number, which you circle like a bud on the tree.

Section 3

Application: Find the Greatest Perfect Square Factor

Property

When finding the prime factorization of a number, we can identify perfect square factors by looking for prime factors that appear an even number of times.
A perfect square factor is formed by taking pairs of identical prime factors.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Systematically Find All Factor Pairs

Property

To find all whole number factor pairs for an area AA, systematically test integers n=1,2,3,…n=1, 2, 3, \dots to see if they divide AA evenly.
If A÷n=mA \div n = m with no remainder, then (n,m)(n, m) is a factor pair.
The process can be stopped when nn becomes greater than mm.

Examples

Section 2

Procedure: Find Prime Factorization with a Factor Tree

Property

To find the prime factorization of a composite number using the tree method:
Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
Step 2. If a factor is prime, that branch is complete. Circle the prime.
Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
Step 4. Write the composite number as the product of all the circled primes.

Examples

  • To factor 36, start with 6⋅66 \cdot 6. Break the first 6 into 2⋅32 \cdot 3 and circle both primes. Break the second 6 into 2⋅32 \cdot 3 and circle both primes. The factorization is 22⋅322^2 \cdot 3^2, or 2⋅2⋅3⋅32 \cdot 2 \cdot 3 \cdot 3.
  • For 28, branch into 4⋅74 \cdot 7. Circle the prime 7. Branch 4 into 2⋅22 \cdot 2 and circle both 2s. The result is 22⋅72^2 \cdot 7, or 2⋅2⋅72 \cdot 2 \cdot 7.
  • To factor 60, you can start with 6⋅106 \cdot 10. Branch 6 into 2⋅32 \cdot 3 and 10 into 2⋅52 \cdot 5. All are prime, so the factorization is 22⋅3⋅52^2 \cdot 3 \cdot 5, or 2⋅2⋅3⋅52 \cdot 2 \cdot 3 \cdot 5.

Explanation

This visual method lets you branch out from a number, splitting it into pairs of factors. You keep branching until every path ends in a prime number, which you circle like a bud on the tree.

Section 3

Application: Find the Greatest Perfect Square Factor

Property

When finding the prime factorization of a number, we can identify perfect square factors by looking for prime factors that appear an even number of times.
A perfect square factor is formed by taking pairs of identical prime factors.

Examples