Prime Factorization
In short: Prime factorization writes a whole number greater than 1 as a product of primes, such as 360 = 2 x 2 x 2 x 3 x 3 x 5, and each number has only one such product.
6th GradeNumbers and Operations6.NS.B.4
What is Prime Factorization?
Every composite number is built from primes, and prime factorization reveals that building plan. The primes act like atoms: they cannot be split further, and multiplying them back together rebuilds the original number.
Precisely, any integer \(n > 1\) equals a product of primes \(p_1 \cdot p_2 \cdots p_k\), and apart from the order of the factors this product is unique. Repeated primes are written with exponents, so \(360 = 2^3 \cdot 3^2 \cdot 5\).
How to spot it
- A question says “write as a product of primes” or asks for the prime factors of a number.
- You need the GCF or LCM of larger numbers and listing factors would take too long.
- A factor tree shows a number branching into smaller numbers.
- A problem asks whether a number is a perfect square, which needs every prime to appear an even number of times.
The method
- Split the number into any two factors greater than 1, starting with a small prime if you can.
- Keep splitting each composite factor until every branch ends in a prime.
- Collect the primes at the ends of the branches and write them in increasing order.
- Use exponents for repeats and multiply to check you recover the original number.
Prime Factorization: solved example
Problem
Write 540 as a product of prime numbers using exponents.
Solution
Split \(540 = 54 \times 10\). Then \(54 = 2 \cdot 27 = 2 \cdot 3 \cdot 3 \cdot 3\) and \(10 = 2 \cdot 5\).
Collect the primes: two 2s, three 3s and one 5. So \(540 = 2^2 \cdot 3^3 \cdot 5\). Check: \(4 \cdot 27 \cdot 5 = 540\).
Try it
Question 1
Find the prime factors of 150.
Show the answer
\(150 = 2 \cdot 75 = 2 \cdot 3 \cdot 25 = 2 \cdot 3 \cdot 5^2\).
Question 2
Is 196 a perfect square? Use its prime factors.
Show the answer
\(196 = 4 \cdot 49 = 2^2 \cdot 7^2\). Every exponent is even, so \(196 = 14^2\) is a perfect square.
Prime Factorization vs. Greatest Common Factor
Prime factorization takes one number apart into primes. The greatest common factor compares two or more numbers and keeps what they share. The first is often the tool you use to find the second, by taking each shared prime to its lowest power.
Keep going
Lessons
Worksheets and quizzes
- Greatest Common Factor and Least Common Multiple – Worksheet
- Greatest Common Factor and Least Common Multiple – Quiz
- Exponents and Order of Operations – Worksheet
- Exponents and Order of Operations – Quiz
Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations