Prime Factorization
Prime factorization is writing a composite number as a product of prime numbers. For example, 100 = 2 x 2 x 5 x 5, and 12 = 2 x 2 x 3. Every composite number has a unique prime factorization, a principle known as the Fundamental Theorem of Arithmetic. Students find prime factorizations using factor trees or repeated division. This skill is taught in Chapter 3 of Saxon Math Course 2 and is critical for 7th grade math because it underlies finding GCF and LCM, simplifying fractions, and working with ratios.
Key Concepts
Property Every composite number can be composed by multiplying two or more prime numbers. When we write a composite number as a product of prime numbers, we are writing the prime factorization of the number.
Examples The prime factorization of $12$ is $2 \cdot 2 \cdot 3$, not $2 \cdot 6$ because $6$ is not a prime number. The prime factorization of $49$ is $7 \cdot 7$. The prime factorization of $100$ is $2 \cdot 2 \cdot 5 \cdot 5$.
Explanation Prime factorization is like finding the secret recipe for a number. You break it down into its most basic, prime ingredients. No matter how you chop it up, you'll always end up with the same set of prime numbers. For example, the number $12$ will always be made of two $2$s and one $3$ in its prime recipe.
Common Questions
What is prime factorization?
Prime factorization is expressing a composite number as a product of prime numbers. For example, the prime factorization of 12 is 2 x 2 x 3.
How do you find the prime factorization of a number?
Use a factor tree: split the number into two factors, then continue splitting non-prime factors until all endpoints are prime. Alternatively, divide by the smallest prime repeatedly.
What is the prime factorization of 100?
100 = 2 x 2 x 5 x 5, or written with exponents, 2^2 x 5^2.
Why is prime factorization important?
It is the basis for finding the greatest common factor and least common multiple, simplifying fractions, and understanding number relationships in algebra.
Can a prime number have a prime factorization?
A prime number is already prime, so its factorization is just itself. For example, the prime factorization of 7 is simply 7.
Is prime factorization part of 7th grade math?
Yes. Saxon Math Course 2 covers prime factorization in Chapter 3, where students use factor trees and division to break down composite numbers.