Methods for Finding the LCM

Just like the GCF, the LCM can be found by three different methods. Pick the one that suits the numbers and the situation.

Method 3 deserves special attention because of what it reveals: LCM(a, b) × GCF(a, b) always equals a × b, no matter what a and b are. This is not a coincidence — it is a direct algebraic identity, and it means that once you (or a computer) can find the GCF quickly using the Euclidean Algorithm, you get the LCM essentially for free, without ever needing to factor either number into primes at all. That combination — fast GCF, free LCM — is exactly why real software never actually lists multiples the way we do by hand.

Method 1 – Listing Multiples

List multiples of the largest number until you find one divisible by the others.

LCM(6, 8) by listing

Multiples of 8: 8, 16, 24, 32...

Is 8 divisible by 6? No. Is 16? No. Is 24? Yes (24÷6=4). LCM = 24

Method 2 – Prime Factorization

Write each number as a product of primes. Take each prime to its highest power across all the numbers. Multiply.

LCM(12, 18, 20)

12 = 2² × 3 18 = 2 × 3² 20 = 2² × 5

Highest powers: 2², 3², 5¹

LCM = 4 × 9 × 5 = 180

Method 3 – Using GCF

Apply the formula: LCM(a,b) = (a × b) ÷ GCF(a,b)

LCM(14, 21)

GCF(14,21): factors of 14 = {1,2,7,14}; factors of 21 = {1,3,7,21}. GCF = 7.

LCM = (14 × 21) ÷ 7 = 294 ÷ 7 = 42

Method Comparison

Method	Best For	Limitation
Listing multiples	Small numbers, quick checks	Slow for large or three-way LCM
Prime factorization	Three or more numbers	Requires factorization first
GCF formula	Exactly two numbers	Does not extend to three+ directly

Key Takeaways

Prime factorization: take the highest power of every prime that appears.
GCF formula is the fastest for two numbers when GCF is easy to find.
For three numbers: find LCM of the first two, then LCM of that result with the third.

Practice: Try a Method

Find the LCM