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Prime
Factorization

Learn to break any number into its prime building blocks and use that power to find the GCF and LCM.

Pre-Algebra

What Is a Prime Number?

A prime number is a whole number greater than \(1\) that has exactly two factors: \(1\) and itself. If a number has more than two factors, it is called composite.

Prime Examples
\(2, 3, 5, 7, 11, 13, 17, 19, 23\) : each divisible only by \(1\) and itself
Composite Examples
\(4, 6, 8, 9, 12, 15, 18, 20\) : each has additional factors beyond \(1\) and itself

What Is Prime Factorization?

Prime factorization is the process of writing a composite number as a product of prime numbers. Every composite number can be broken down in exactly one way (ignoring order).

We call \(2^2 \times 3 \times 5\) the prime factorization of \(60\). Every factor in the product is prime, and multiplying them all together returns the original number.

Why Does It Matter?

Simplify Fractions
Finding the GCF lets you reduce fractions to lowest terms quickly.
Add Unlike Fractions
The LCM gives you the least common denominator so you can add or subtract.
Real-World Uses
Scheduling, cryptography, and number theory all rely on prime factorization.

The Fundamental Theorem of Arithmetic

This theorem guarantees that no matter how you break a number down, you always arrive at the same set of primes. For example:

\(36\)
\(2^2 \times 3^2\) : the only prime factorization
\(100\)
\(2^2 \times 5^2\) : the only prime factorization

The Factor Tree Method

Start with the number at the top. Split it into any two factors. Keep splitting until every leaf is prime.

60 2 30 5 6 2 3 Primes = green circles

Prime vs Composite

Prime Composite
Factors Exactly 2: \(1\) and itself More than 2
Smallest \(2\) \(4\)
Examples \(2, 3, 5, 7, 11\) \(4, 6, 9, 12, 15\)
Can be factored further? No Yes
Role in factorization Building blocks (leaves) Intermediate nodes

How to Factor Any Number

1
Is it prime? Check divisibility starting from the smallest prime (\(2\)). If no prime up to \(\sqrt{n}\) divides it, the number is prime and you are done.
2
Divide by the smallest prime that goes in evenly. Write the quotient beside it on the factor tree.
3
Repeat on every composite quotient until every branch ends in a prime.
4
Collect all the prime leaves and write the product using exponents.

Example: Factor \(72\)

72 2 36 2 18 2 9 3 3

Example: Factor \(180\)

Using repeated division (ladder method):

1
\(180 \div 2 = 90\)
2
\(90 \div 2 = 45\)
3
\(45 \div 3 = 15\)
4
\(15 \div 3 = 5\)
5
\(5\) is prime: stop.

Finding the GCF

The Greatest Common Factor is the product of the common prime factors, each taken to its lowest power.

1
Write the prime factorization of each number.
2
Identify the primes that appear in both factorizations.
3
Take the lower exponent for each shared prime.
4
Multiply those together.

Finding the LCM

The Least Common Multiple is the product of all unique prime factors from both numbers, each taken to its highest power.

1
Write the prime factorization of each number.
2
List every prime that appears in either factorization.
3
Take the higher exponent for each prime.
4
Multiply those together.

GCF and LCM Visualized

72 180 2 extra factor of 2 Shared (GCF)
\(2^2 \times 3^2\)
\(= 36\)
 5 extra factor of 5 LCM = all factors at highest power
\(2^3 \times 3^2 \times 5 = 360\)

Example: \(\text{GCF}\) and \(\text{LCM}\) of \(48\) and \(60\)

\(48\)
\(48 = 2^4 \times 3\)
\(60\)
\(60 = 2^2 \times 3 \times 5\)
Prime Power in 48 Power in 60 For GCF (min) For LCM (max)
\(2\) \(4\) \(2\) \(2\) \(4\)
\(3\) \(1\) \(1\) \(1\) \(1\)
\(5\) \(0\) \(1\) \(0\) \(1\)

Common Mistakes

✕ Stopping Too Early
Writing \(12 = 4 \times 3\) and stopping. \(4\) is composite: \(12 = 2^2 \times 3\). Every factor must be prime.
✕ Swapping GCF/LCM Rules
Using the highest power for GCF or the lowest for LCM. GCF uses min, LCM uses max.
✕ Calling 1 Prime
Including \(1\) in a factorization. \(1\) is not prime and should never appear as a factor in the product.

Summary Table

GCF LCM
Stands for Greatest Common Factor Least Common Multiple
Which primes? Only shared primes All primes from both
Which exponents? Minimum power Maximum power
Result is... \( \le \) either number \( \ge \) either number
Used for Simplifying fractions Common denominators
Relationship \(\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b\)

Question 1 of 5

What is the prime factorization of \(90\)?

Question 2 of 5

Given \(36 = 2^2 \times 3^2\) and \(48 = 2^4 \times 3\), what is \(\text{GCF}(36, 48)\)?

Question 3 of 5

What is \(\text{LCM}(12, 18)\)?

Question 4 of 5

Which statement about the number \(1\) is true?

Question 5 of 5

If \(\text{GCF}(a, b) = 15\) and \(\text{LCM}(a, b) = 180\), what is \(a \times b\)?