How to Find the Least Common Multiple (LCM)

The least common multiple (LCM) of two or more integers is the smallest positive integer divisible by all of them. This page explains three methods — listing multiples, prime factorization, and the GCD formula — with fully worked examples.

Definition

LCM(a, b) is the smallest positive integer that is a multiple of both a and b. It is also called the lowest common multiple or smallest common multiple.

ℹLCM(12, 18) = 36. Multiples of 12: 12, 24, 36, 48… Multiples of 18: 18, 36, 54… The smallest shared value is 36.

Method 1 — Listing Multiples

List multiples of each number until you find the first one they share. Practical only for small numbers.

Number	Multiples
12	12, 24, 36, 48, 60, 72, …
18	18, 36, 54, 72, …
First common multiple	36
LCM	36

Method 2 — Prime Factorization

Write each number as a product of prime powers. The LCM is the product of all primes, each raised to the maximum exponent.

lcm (a, b) = p prime \prod p^{m a x (e_{p} (a), e_{p} (b))}

Example: LCM(360, 756)

360 = 2^{3} \times 3^{2} \times 5, 756 = 2^{2} \times 3^{3} \times 7

Prime	Exponent in 360	Exponent in 756	max → used in LCM
2	3	2	3
3	2	3	3
5	1	0	1
7	0	1	1

lcm (360, 756) = 2^{3} \times 3^{3} \times 5 \times 7 = 8 \times 27 \times 5 \times 7 = 7560

Method 3 — GCD Formula (Most Efficient)

For two numbers, use the identity that connects LCM and GCD:

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

Worked Example: LCM(252, 105)

Step 1 — Find GCD(252, 105) using the Euclidean algorithm:

Step	a	b	a = b × q + r	Remainder r
1	252	105	252 = 105 × 2 + 42	42
2	105	42	105 = 42 × 2 + 21	21
3	42	21	42 = 21 × 2 + 0	0

GCD(252, 105) = 21.

Step 2 — Apply the LCM formula:

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

Worked Example: LCM(8, 15)

GCD(8, 15) = 1 (coprime). Therefore:

lcm (8, 15) = \frac{8 \times 15}{1} = 120

💡When two numbers are coprime (GCD = 1), their LCM is simply their product.

LCM of More Than Two Numbers

Apply LCM pairwise using the associative property:

lcm (a, b, c) = lcm (lcm (a, b), c)

Example: LCM(4, 6, 10) → LCM(LCM(4, 6), 10) = LCM(12, 10) = 60.

Key Properties of LCM

Property	Statement	Example
Commutative	LCM(a, b) = LCM(b, a)	LCM(6, 10) = LCM(10, 6) = 30
Identity	LCM(a, 1) = a	LCM(7, 1) = 7
GCD × LCM	GCD(a, b) × LCM(a, b) = a × b	6 × 60 = 12 × 30 = 360
Divisor relationship	If a divides b, then LCM(a, b) = b	LCM(6, 18) = 18

Applications

Adding and subtracting fractions: use LCM as the common denominator. Example: 1/4 + 1/6 → LCD = LCM(4, 6) = 12 → 3/12 + 2/12 = 5/12.
Scheduling problems: if event A repeats every 12 days and event B every 18 days, they next coincide in LCM(12, 18) = 36 days.
Gear and wheel synchronization: LCM gives the number of rotations before a pattern repeats.
Music theory: LCM determines when two rhythmic patterns realign.

💡The LCM identity GCD(a, b) × LCM(a, b) = a × b is a powerful shortcut: if you know the GCD, you get the LCM for free, and vice versa.

LCM Calculator — Calculate LCM with step-by-step GCD method output
LCM Formula Reference — Formula, variables, and worked examples

Frequently Asked Questions

What is the LCM of two consecutive integers?

Consecutive integers are always coprime (GCD = 1), so LCM(n, n+1) = n × (n+1). For example, LCM(7, 8) = 56.

Can LCM be smaller than both numbers?

No. LCM(a, b) ≥ max(a, b) always. The LCM is at least as large as the larger of the two numbers.

How is LCM used to add fractions?

To add fractions with different denominators, find the LCM of the denominators (this is the lowest common denominator, LCD). Convert both fractions to the LCD, then add numerators. Example: 1/6 + 1/4 → LCD = LCM(6, 4) = 12 → 2/12 + 3/12 = 5/12.

What is LCM(a, a)?

LCM(a, a) = a. A number is always a multiple of itself, so the smallest shared multiple of a number with itself is the number.

Is there a formula for LCM of three numbers using prime factorization?

Yes: LCM(a, b, c) = product of all primes appearing in any factorization, each raised to the highest exponent seen. Example: LCM(4, 6, 10) = LCM(2², 2×3, 2×5) = 2² × 3 × 5 = 60.