LCM vs GCF: What's the Difference? (With Examples)

The GCF is the largest number that divides into your numbers; the LCM is the smallest number your numbers divide into. One looks down to shared factors, the other looks up to shared multiples.

Mixing them up is one of the most common math mistakes. Here is a simple way to keep them straight, plus the elegant formula that connects the two.

Definitions at a glance

• GCF (Greatest Common Factor) = the biggest number that divides all inputs. Also called GCD or HCF.
• LCM (Least Common Multiple) = the smallest number that all inputs divide into.

For 12 and 18: GCF = 6, LCM = 36.

A memory trick

Factor → fewer/smaller (factors fit inside the number). Multiple → more/bigger (multiples grow beyond it). So the GCF is small and the LCM is large.

When to use each

• Use the GCF to simplify fractions, split items into equal groups, or reduce a ratio. Try the GCD Calculator.
• Use the LCM to add fractions (common denominator) or align repeating cycles. Try the LCM Calculator.

The formula that links them

For any two numbers, LCM(a, b) × GCF(a, b) = a × b. For 12 and 18: 36 × 6 = 216 = 12 × 18. Knowing one gives you the other instantly.