Two related number-theory questions answered together, since they’re commonly taught and searched as a pair.
How it works
The GCF is found with the Euclidean algorithm; the LCM is then found by multiplying the two numbers together and dividing by the GCF.
What this does not include
This does not include prime factorization steps shown explicitly — for just the GCF alone with a simpler page, use this site’s GCF calculator instead.
How to use this calculator
- Enter both whole numbers.
A worked example
48 and 18: the Euclidean algorithm gives a GCF of 6. LCM = (48 × 18) ÷ 6 = 144.
21 and 6: GCF is 3, so LCM = (21 × 6) ÷ 3 = 42.
What the terms mean
| Term | Meaning |
|---|---|
| GCF | Greatest common factor — the largest number dividing evenly into both |
| LCM | Least common multiple — the smallest number both divide into evenly |
Edge cases worth knowing
Two coprime numbers (sharing no common factor but 1) have a GCF of exactly 1 — a valid, common result, not an error. Their LCM in that case is simply their product.
Any number’s GCF and LCM with itself are that number — GCF(7, 7) = 7 and LCM(7, 7) = 7.
Frequently asked questions
Why does multiplying by GCF and dividing give the LCM?
Because the product of two numbers always equals their GCF times their LCM — a well-known number-theory identity, rearranged here to solve for LCM directly.
What’s a practical use for LCM?
Finding a common denominator when adding fractions, or figuring out when two repeating events (like two bus schedules) will next align.
Can GCF ever be larger than LCM?
No — GCF is always less than or equal to the smaller of the two numbers, while LCM is always greater than or equal to the larger one.
