Least Common Multiple (LCM) - Methods & Examples

The least common multiple (LCM) of two or more positive integers is the smallest positive number they all share as a multiple. For example, $\mathrm{LCM}(6,8) = 24$, because $24$ is a multiple of both and nothing smaller works. This is the idea behind common denominators, repeating schedules, and any question about when two patterns line up again.

The formula and its key tool

The most compact rule connects the LCM to the greatest common divisor. For two positive integers $a$ and $b$,

The prime-factorization method is the other workhorse: write each number as a product of primes, then keep every prime that appears, using the largest exponent seen for each prime.

Keep one contrast straight throughout: a factor divides a number, while a multiple is produced by multiplying it. LCM is about multiples, not factors.

Why the methods work

The GCD formula holds because $a \cdot b$ counts every shared prime power twice. Dividing by $\mathrm{GCD}(a,b)$ removes exactly the overlap, leaving the smallest number that still contains both.

The prime-power rule works for the same reason from the other direction. To be divisible by both numbers, a multiple must include each prime to at least the highest power either number demands. Taking the larger exponent supplies exactly that — no more, no less — which is why it produces the least common multiple rather than just any common multiple.

Worked example: the LCM of $12$ and $18$

Use prime factorization:

Keep each prime with the larger exponent:

• For $2$, the larger exponent is $2$
• For $3$, the larger exponent is $2$

So:

Check directly:

• $36 \div 12 = 3$
• $36 \div 18 = 2$

Both divide evenly, so $36$ is a common multiple, and the prime-power construction guarantees it is the least one.

This shows up constantly in fraction addition. The denominators in $\frac{1}{6} + \frac{1}{8}$ have LCM $24$, so

You also use LCM when two repeating events occur every $m$ and $n$ units and you want the first time they coincide.

Practice with two checks

Find the LCM of $15$ and $20$ two ways: by listing multiples and by prime factorization. Both routes should give $60$. After you land on an answer, run two checks — is it divisible by each original number, and is there any smaller positive common multiple? For the prime-power method, the second check is baked into the method itself.

Common pitfalls

• Mixing up LCM and GCD. Smallest shared multiple means LCM; greatest shared factor means GCD.
• Stopping at a common multiple that is not the least. For $6$ and $8$, both $24$ and $48$ are common multiples, but only $24$ is the LCM.
• Applying the "larger exponent" rule without factoring first. That rule only makes sense once the numbers are written as prime factorizations of positive integers.