Lesson 9: Prime Numbers

New Concept

Prime numbers are counting numbers greater than 1 that have exactly two different counting number factors, the number itself and 1.

What’s next

This lesson is your starting point. Soon, we'll use worked examples with factor trees and divisibility tests to find the prime components of any number.

Property

Prime numbers are counting numbers greater than 1 that have exactly two different counting number factors, the number itself and 1.

Examples

• The number 7 is prime because its only factors are 1 and 7. You can only form a $1 \times 7$ rectangle with 7 tiles.
• The number 13 is prime. Its only factor pair is $1 \times 13$, meaning it cannot be divided evenly by any other counting numbers.
• The number 2 is the only even prime number, because all other even numbers are divisible by 2, giving them an extra factor.

Explanation

Think of prime numbers as the ultimate loners! If you have a prime number of tiles, like 7, you can only make one boring rectangle (a 7-by-1). They refuse to be arranged in any other interesting way because they only have two factors: themselves and 1. It’s their way or the highway, making them the basic building blocks for all other numbers.

Property

Counting numbers with more than two factors are composite numbers. Composite numbers are so named because they are composed of two or more prime factors.

Examples

• The number 6 is composite. Its factors are 1, 2, 3, and 6. It can be modeled as a $1 \times 6$ rectangle or a $2 \times 3$ rectangle.
• The number 12 is composite. You can form three different rectangles with 12 squares: $1 \times 12$, $2 \times 6$, and $3 \times 4$.
• The number 9 is composite because its factors are 1, 3, and 9. It can form a $1 \times 9$ rectangle and a $3 \times 3$ square.

Explanation

Composite numbers are the social butterflies of the number world! Unlike lonely primes, they have multiple factor friends. If you have a composite number of tiles, like 6, you can arrange them in different rectangular groups, such as a 6-by-1 or a 3-by-2. They are literally composed of smaller prime factors, making them much more flexible and fun!

Property

We write the prime factorization of a composite number by writing the number as a product of prime numbers.

Examples

• Using a factor tree for 36: $36 \rightarrow 4 \times 9 \rightarrow (2 \times 2) \times (3 \times 3)$. The prime factorization is $2 \cdot 2 \cdot 3 \cdot 3$.
• Using division for 60: $60 \div 2 = 30$; then $30 \div 2 = 15$; then $15 \div 3 = 5$. The prime factors are $2 \cdot 2 \cdot 3 \cdot 5$.
• The prime factorization of 45 is found by division: $45 \div 5 = 9$, and $9 \div 3 = 3$. So, the final product is $3 \cdot 3 \cdot 5$.

Explanation

Think of prime factorization as revealing a number's secret identity! Every composite number has a unique set of prime factors that multiply together to create it. A factor tree is like a family tree for numbers, showing the prime ancestors at the very bottom. It's like breaking down a cookie to find its essential ingredients—but with numbers instead of chocolate chips!