GCF and LCM: How to Find Them

The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) show up everywhere in math — from simplifying fractions to finding common denominators to solving word problems. They sound similar, but they answer very different questions. This tutorial covers both, with two methods for each and plenty of examples.

What is the GCF?

The Greatest Common Factor (also called the Greatest Common Divisor, or GCD) of two numbers is the largest number that divides evenly into both of them.

For example, the GCF of 12 and 18 is 6 because 6 is the biggest number that goes into both 12 and 18 with no remainder.

What is the LCM?

The Least Common Multiple of two numbers is the smallest positive number that is a multiple of both.

For example, the LCM of 4 and 6 is 12 because 12 is the first number that appears in both the 4-times table and the 6-times table.

Method 1: Listing

The most straightforward approach — write out lists and look for what they share.

Listing method for GCF

List all factors of each number, then pick the largest one they have in common.

GCF Example 1: GCF of 12 and 18

GCF(12, 18) = 6

GCF Example 2: GCF of 20 and 35

GCF(20, 35) = 5

Listing method for LCM

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

LCM Example 1: LCM of 4 and 6

LCM(4, 6) = 12

LCM Example 2: LCM of 5 and 8

LCM(5, 8) = 40

Listing works well for small numbers, but for larger ones it gets tedious. That's where prime factorization comes in.

Method 2: Prime factorization

Break each number into its prime factors, then use those primes to build the GCF or LCM.

Prime factorization for GCF

Multiply together the shared primes, using the lowest power of each.

GCF Example 3: GCF of 24 and 36

GCF(24, 36) = 12

GCF Example 4: GCF of 45 and 75

GCF(45, 75) = 15

Prime factorization for LCM

Multiply together all primes that appear in either number, using the highest power of each.

LCM Example 3: LCM of 12 and 18

LCM(12, 18) = 36

LCM Example 4: LCM of 8 and 14

LCM(8, 14) = 56

The GCF × LCM shortcut

There's a handy relationship between the GCF and LCM of two numbers:

This means if you already know the GCF, you can find the LCM (or vice versa) with a simple division:

Example: Find the LCM of 12 and 18 using the shortcut.

Same answer as the prime factorization method — 36. ✓

When to use GCF vs. LCM

Knowing which one you need is often the hardest part of a problem. Here's a quick guide:

• Use GCF when simplifying fractions. Divide both the numerator and denominator by the GCF to reduce to lowest terms. For example, to simplify 24/36: GCF = 12, so 24/36 = 2/3.
• Use GCF when splitting things into equal groups. "What's the largest number of identical groups you can make?"
• Use LCM when finding common denominators. To add 1/4 + 1/6, you need a common denominator — the LCM of 4 and 6 is 12.
• Use LCM for repeating-event problems. "Two buses leave at different intervals. When will they leave at the same time again?"

More worked examples

GCF of 30 and 42

GCF(30, 42) = 6

LCM of 9 and 15

LCM(9, 15) = 45

GCF of 48 and 64

GCF(48, 64) = 16

LCM of 10 and 12

LCM(10, 12) = 60

Common mistakes

Confusing GCF and LCM: The GCF is always less than or equal to both numbers. The LCM is always greater than or equal to both numbers. If your "GCF" is larger than one of the numbers, you've actually found the LCM (or made an error).

Forgetting to use all primes for LCM: When building the LCM, you must include primes from both numbers, not just the shared ones. A prime that only appears in one number still goes into the LCM at its highest power.

Using highest powers for GCF: For the GCF, use the lowest power of each shared prime. Using the highest power gives the LCM instead.

Stopping the prime factorization too early: Make sure each factor is actually prime. 4 is not prime (it's 2 × 2), and 9 is not prime (it's 3 × 3). Keep breaking factors down until every piece is prime.

Quick tips

• GCF ≤ both numbers; LCM ≥ both numbers
• If two numbers share no common factor besides 1 (they're "coprime"), the GCF is 1 and the LCM is their product
• If one number is a multiple of the other, the GCF is the smaller number and the LCM is the larger number
• Always double-check with: GCF × LCM = product of the two numbers

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