Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 3: Number Theory

Lesson 4: Prime Factorization

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students learn how to find the prime factorization of any positive integer by breaking it down into a product of prime factors, including the use of factor trees and exponential notation. The lesson covers key vocabulary such as prime factors, composite numbers, and the Fundamental Theorem of Arithmetic, which states that every positive integer has a unique prime factorization. Students also explore how prime factorizations relate to perfect squares by examining the exponents in a number's prime factorization.

Section 1

Prime factorization

Property

The prime factorization of a number is the product of prime numbers that equals the number. Every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes.

Examples

The prime factorization of 30 is $2 \cdot 3 \cdot 5$ , which cannot be simplified further in exponential form.
For the number 77, the prime factorization is simply $7 \cdot 11$ , since both are prime numbers.
The prime factorization for 84 is $2 \cdot 2 \cdot 3 \cdot 7$ , or $2^2 \cdot 3 \cdot 7$ .

Explanation

Think of prime factorization as finding a number's unique recipe. Each composite number is built from a special combination of prime number ingredients that, when multiplied together, give you back the original number.

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 $6 \cdot 6$ . Break the first 6 into $2 \cdot 3$ and circle both primes. Break the second 6 into $2 \cdot 3$ and circle both primes. The factorization is $2^2 \cdot 3^2$ , or $2 \cdot 2 \cdot 3 \cdot 3$ .
For 28, branch into $4 \cdot 7$ . Circle the prime 7. Branch 4 into $2 \cdot 2$ and circle both 2s. The result is $2^2 \cdot 7$ , or $2 \cdot 2 \cdot 7$ .
To factor 60, you can start with $6 \cdot 10$ . Branch 6 into $2 \cdot 3$ and 10 into $2 \cdot 5$ . All are prime, so the factorization is $2^2 \cdot 3 \cdot 5$ , or $2 \cdot 2 \cdot 3 \cdot 5$ .

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Prime factorization

Property

Examples

The prime factorization of 30 is $2 \cdot 3 \cdot 5$ , which cannot be simplified further in exponential form.
For the number 77, the prime factorization is simply $7 \cdot 11$ , since both are prime numbers.
The prime factorization for 84 is $2 \cdot 2 \cdot 3 \cdot 7$ , or $2^2 \cdot 3 \cdot 7$ .

Explanation

Section 2

Procedure: Find Prime Factorization with a Factor Tree

Property

Examples

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

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.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Prime factorization

Property

Examples

The prime factorization of 30 is $2 \cdot 3 \cdot 5$ , which cannot be simplified further in exponential form.
For the number 77, the prime factorization is simply $7 \cdot 11$ , since both are prime numbers.
The prime factorization for 84 is $2 \cdot 2 \cdot 3 \cdot 7$ , or $2^2 \cdot 3 \cdot 7$ .

Explanation

Section 2

Procedure: Find Prime Factorization with a Factor Tree

Property

Examples

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

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.