LCM Formula – Least Common Multiple Calculation

The complete LCM formula using the GCD method and prime factorization: variable definitions, step-by-step method, and three fully worked examples.

Formula

lcm (a, b) = \frac{a \times b}{g cd ( a , b )}

The least common multiple of two positive integers equals their product divided by their greatest common divisor. For more than two numbers, apply the formula pairwise.

Variables

Symbol	Name	Description	Unit
a	First integer	A positive integer	—
b	Second integer	A positive integer	—
GCD	Greatest Common Divisor	GCD(a, b) — the largest integer that divides both a and b; computed with the Euclidean algorithm	—
LCM	Least Common Multiple	The smallest positive integer divisible by both a and b	—

How to Use

Find GCD(a, b) using the Euclidean algorithm.
Compute LCM(a, b) = (a × b) ÷ GCD(a, b).
For more than two numbers: LCM(a, b, c) = LCM(LCM(a, b), c).

Examples

1. LCM(12, 18)

Step 1 — find GCD(12, 18) using the Euclidean algorithm.

Step	a	b	a = b × q + r	Remainder r
1	12	6	12 = 6 × 2 + 0	0

GCD(12, 18): 18 = 12 × 1 + 6, then 12 = 6 × 2 + 0 → GCD = 6.

lcm (12, 18) = \frac{12 \times 18}{6} = \frac{216}{6} = 36

ℹVerification: 36 ÷ 12 = 3 ✓ 36 ÷ 18 = 2 ✓ No smaller common multiple exists.

2. LCM(252, 105)

Step 1 — GCD(252, 105) via Euclidean algorithm:

Step	a	b	Remainder r
1	252	105	42
2	105	42	21
3	42	21	0

GCD = 21.

lcm (252, 105) = \frac{252 \times 105}{21} = \frac{26 , 460}{21} = 1, 260

ℹ1,260 ÷ 252 = 5 ✓ 1,260 ÷ 105 = 12 ✓

3. LCM(4, 6, 10) — three numbers

Apply LCM pairwise. First compute LCM(4, 6), then LCM(result, 10).

g cd (4, 6) = 2 ⟹ lcm (4, 6) = \frac{4 \times 6}{2} = 12

g cd (12, 10) = 2 ⟹ lcm (12, 10) = \frac{12 \times 10}{2} = 60

lcm (4, 6, 10) = 60

💡Verify using prime factorization: 4 = 2², 6 = 2 × 3, 10 = 2 × 5. LCM = 2² × 3 × 5 = 60. ✓

Use the Calculator — Interactive calculator for this formula
Read the Notes — Step-by-step explanation with worked examples