GCF and LCM

The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers. The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. You use the GCF to simplify fractions and the LCM to find common denominators — two of the most frequent tasks in arithmetic.

Greatest Common Factor (GCF)

The GCF is also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

Method 1: List the Factors

List all factors of each number, then find the largest one they share.

Example 1: Find the GCF of 18 and 24

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Common factors: 1, 2, 3, 6

GCF = 6

Method 2: Prime Factorization

Find the prime factorization of each number. The GCF is the product of the shared prime factors, each taken to the lowest power it appears.

Example 2: Find the GCF of 36 and 60

$36 = 2^2 \times 3^2$ $60 = 2^2 \times 3 \times 5$

Shared primes: 2 and 3.

• $2$: lowest power is $2^2$
• $3$: lowest power is $3^1$

$\text{GCF} = 2^2 \times 3 = 4 \times 3 = 12$

Example 3: Find the GCF of 45, 60, and 75

$45 = 3^2 \times 5$ $60 = 2^2 \times 3 \times 5$ $75 = 3 \times 5^2$

Shared primes: 3 and 5.

• $3$: lowest power is $3^1$
• $5$: lowest power is $5^1$

$\text{GCF} = 3 \times 5 = 15$

Least Common Multiple (LCM)

Method 1: List the Multiples

List multiples of each number until you find the first one they share.

Example 4: Find the LCM of 4 and 6

• Multiples of 4: 4, 8, 12, 16, 20, 24…
• Multiples of 6: 6, 12, 18, 24…

LCM = 12

Method 2: Prime Factorization

Find the prime factorization of each number. The LCM is the product of all prime factors, each taken to the highest power it appears.

Example 5: Find the LCM of 12 and 18

$12 = 2^2 \times 3$ $18 = 2 \times 3^2$

All primes: 2 and 3.

• $2$: highest power is $2^2$
• $3$: highest power is $3^2$

$\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36$

Example 6: Find the LCM of 8, 12, and 15

$8 = 2^3$ $12 = 2^2 \times 3$ $15 = 3 \times 5$

All primes: 2, 3, 5.

• $2$: highest power is $2^3$
• $3$: highest power is $3^1$
• $5$: highest power is $5^1$

$\text{LCM} = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$

GCF and LCM Relationship

For any two numbers $a$ and $b$:

$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Example 7: Verify with 12 and 18

$\text{GCF}(12, 18) = 6$ and $\text{LCM}(12, 18) = 36$

$6 \times 36 = 216 = 12 \times 18 \checkmark$

This formula is a quick way to find the LCM if you already know the GCF:

$\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}$

When to Use GCF vs. LCM

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• GCF: largest factor shared by two or more numbers — use lowest powers of shared primes
• LCM: smallest multiple shared by two or more numbers — use highest powers of all primes
• GCF simplifies fractions; LCM finds common denominators
• The relationship: $\text{GCF} \times \text{LCM} = a \times b$
• Both the listing method and prime factorization method work — use whichever is faster for the numbers

Return to Arithmetic for more foundational math topics.