What is the LCM of 12, 18 and 30?

180 — the smallest number that 12, 18 and 30 all divide into.

Are GCF, HCF and GCD the same?

Yes. Greatest common factor, highest common factor and greatest common divisor are three names for the same thing.

How does Euclid's algorithm work?

Replace the pair (a, b) with (b, a mod b) until the remainder is 0. For 84 and 60: 84 = 1×60 + 24, 60 = 2×24 + 12, 24 = 2×12 + 0, so the GCF is 12.

What is the relationship between LCM and GCF?

For two numbers, LCM × GCF = a × b. So LCM(48, 180) = 48 × 180 ÷ 12 = 720.

Can I use decimals or negative numbers?

No. LCM and GCF are defined here for positive whole numbers.

Math · Free tool

LCM & GCF Calculator

Least common multiple and greatest common factor of up to 20 numbers, with Euclid's steps.

The greatest common factor (GCF) — also called the highest common factor (HCF) or greatest common divisor (GCD) — is the largest whole number that divides every number in a set exactly. The least common multiple (LCM) is the smallest whole number that every number in the set divides into. They answer opposite questions: GCF breaks things into the largest equal groups; LCM finds when repeating cycles line up again.

This calculator handles 2 to 20 positive whole numbers. It uses Euclid's algorithm, the 2,300-year-old method that repeatedly replaces the larger number with the remainder of a division, and shows those steps for the first pair so you can follow the working. The LCM is then built from the GCF using LCM(a, b) = a × b ÷ GCF(a, b).

You will meet both in fraction arithmetic (the LCM of denominators is the lowest common denominator; the GCF simplifies a fraction), in scheduling problems, and in tiling and packing questions.

By DailyCalc Editorial Team · Updated 2026-10-03 · 5-min read · 3 sources cited

● LCM & GCF calculator

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Euclid's algorithm and the LCM formula

GCF(a, b) = GCF(b, a mod b), until the remainder is 0; LCM(a, b) = a × b ÷ GCF(a, b)

Divide the larger number by the smaller and keep the remainder. Repeat with the smaller number and that remainder. When the remainder reaches 0, the last non-zero divisor is the GCF. For more than two numbers, apply the rule pairwise: GCF(a, b, c) = GCF(GCF(a, b), c), and the same for LCM.

a, b: Positive whole numbers.
a mod b: The remainder when a is divided by b.

Worked example: 12, 18 and 30

Find both the GCF and the LCM:

GCF(12, 18): 18 = 1 × 12 + 6; 12 = 2 × 6 + 0 → GCF = 6.
GCF(6, 30): 30 = 5 × 6 + 0 → GCF of all three = 6.
LCM(12, 18) = 12 × 18 ÷ 6 = 36.
LCM(36, 30): GCF(36, 30) = 6, so LCM = 36 × 30 ÷ 6 = 180.
Check: 180 ÷ 12 = 15, 180 ÷ 18 = 10, 180 ÷ 30 = 6 — all whole numbers ✓.

Result: GCF(12, 18, 30) = 6 and LCM(12, 18, 30) = 180.

How to use this calculator

One box, any separator.

Step 1: Enter 2–20 whole numbers

Separate them with commas or spaces, for example 12, 18, 30. Decimals and negative numbers are not accepted.

Step 2: Calculate

You get the GCF, the LCM and Euclid's steps for the first two numbers.

Where LCM and GCF come up

Everyday and classroom problems:

Adding fractions

To add 5/12 + 7/18, use the LCM of 12 and 18 (36) as the common denominator: 15/36 + 14/36 = 29/36.

Simplifying fractions

Divide top and bottom by their GCF: 84/60 has GCF 12, so it simplifies to 7/5.

Scheduling

Buses every 12 and 18 minutes leave together again after LCM(12, 18) = 36 minutes.

Equal groups

48 pens and 180 pencils packed into identical kits with nothing left over: GCF(48, 180) = 12 kits.

Gears and cycles

Gear teeth counts and repeating patterns realign after their LCM.

Pro tips

Shortcuts worth knowing:

For any two numbers, GCF × LCM = a × b — a quick check on your answer.
If one number divides the other, the smaller is the GCF and the larger is the LCM.
Two consecutive whole numbers always have a GCF of 1.
Euclid's algorithm is much faster than listing factors for large numbers.

Common mistakes

Where students slip:

Multiplying instead of finding the LCM

12 × 18 = 216 is a common multiple, but the least common multiple is 36.

Mixing up GCF and LCM

The GCF can be no larger than the smallest number; the LCM is at least as large as the largest.

Applying the two-number formula to three at once

a × b × c ÷ GCF(a, b, c) is not the LCM. Work pairwise.

Stopping Euclid too early

Keep going until the remainder is exactly 0; the GCF is the last non-zero divisor.

Sources & references

The formula, thresholds and guidance on this page were checked against these primary sources. Written and reviewed by DailyCalc Editorial Team; last updated 2026-10-03. Spotted an error? Tell us.

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