Prime Factorization, GCF, and LCM

By the end of this lesson, you will be able to find the prime factorization of a number, compute the GCF and LCM, and apply them to simplify fractions and solve problems.

Prime and Composite Numbers

Prime number: exactly two factors (1 and itself). Examples: 2, 3, 5, 7, 11, 13, ...

Composite number: more than two factors. Examples: 4, 6, 8, 9, 10, ...

1 is neither prime nor composite.

2 is the only even prime.

Prime Factorization

36
├── 4 × 9
│   ├── 2 × 2
│   └── 3 × 3

Every composite number can be written as a product of prime numbers — this is its prime factorization (unique by the Fundamental Theorem of Arithmetic).

Method — Factor Tree:

$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

Method — Repeated Division (Cake Method):

Divide by the smallest prime repeatedly:

$\frac{72}{2} = 36 \to \frac{36}{2} = 18 \to \frac{18}{2} = 9 \to \frac{9}{3} = 3 \to \frac{3}{3} = 1$

$72 = 2^3 \times 3^2$

Greatest Common Factor (GCF)

GCF(a, b): the largest number that divides both a and b evenly.

Method — Prime Factorization:

1. Factor each number into primes
2. Circle the common prime factors (including multiplicity)
3. Multiply the common factors

Example: GCF(48, 60)

$48 = 2^4 \times 3$; $60 = 2^2 \times 3 \times 5$

Common factors: $2^2$ and $3^1$

$\text{GCF}(48, 60) = 2^2 \times 3 = 12$

Use GCF to: simplify fractions ($\frac{48}{60} = \frac{4}{5}$ after dividing by 12), divide items into equal groups.

Least Common Multiple (LCM)

LCM(a, b): the smallest positive number divisible by both a and b.

Method — Prime Factorization:

1. Factor each number into primes
2. Take the highest power of each prime that appears
3. Multiply them