Least Common Multiple (LCM) - Methods & Examples

The least common multiple, or LCM, is the smallest positive number that two or more positive integers share as a multiple. For example, the LCM of $6$ and $8$ is $24$ because $24$ is a multiple of both numbers, and no smaller positive number works.

This is the idea you usually need for common denominators, repeating schedules, and questions that ask when two patterns line up again.

What LCM Means

A multiple of $6$ is any number of the form $6k$ for a positive integer $k$: $6, 12, 18, 24, \dots$

A multiple of $8$ is any number of the form $8k$: $8, 16, 24, 32, \dots$

The first positive number that appears in both lists is $24$, so:

$$\mathrm{LCM}(6,8) = 24$$

It helps to keep one contrast in mind:

• A factor divides a number.
• A multiple is produced by multiplying a number.

LCM is about multiples, not factors.

Three Reliable Ways To Find LCM

1. List Multiples

This works well for small numbers.

For $4$ and $10$:

• Multiples of $4$: $4, 8, 12, 16, 20, \dots$
• Multiples of $10$: $10, 20, 30, \dots$

The first common multiple is $20$, so the LCM is $20$.

2. Use Prime Factorization

This is often the clearest method for larger positive integers.

Write each number as a product of primes, then keep every prime that appears, using the largest exponent that appears for each prime.

3. Use The GCD Relationship

For two positive integers $a$ and $b$,

$$\mathrm{LCM}(a,b) = \frac{a \cdot b}{\mathrm{GCD}(a,b)}$$

This method is efficient if you already know the greatest common divisor. The condition matters: this formula is used for positive integers.

Worked Example: Find The LCM Of $12$ And $18$

Use prime factorization:

$$12 = 2^2 \cdot 3$$ $$18 = 2 \cdot 3^2$$

To build the LCM, keep each prime with the larger exponent:

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

So:

$$\mathrm{LCM}(12,18) = 2^2 \cdot 3^2 = 36$$

Check it directly:

• $36 \div 12 = 3$
• $36 \div 18 = 2$

So $36$ is a common multiple. The prime factor method gives the least one because it uses exactly the prime powers needed to include both numbers.

When LCM Is Used

LCM is useful when a problem asks for a shared cycle or a shared denominator.

One common example is adding fractions:

$$\frac{1}{6} + \frac{1}{8}$$

The denominators $6$ and $8$ have LCM $24$, so $24$ is a convenient common denominator:

$$\frac{1}{6} = \frac{4}{24}, \qquad \frac{1}{8} = \frac{3}{24}$$

Then:

$$\frac{1}{6} + \frac{1}{8} = \frac{7}{24}$$

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

Common Mistakes

Mixing Up LCM And GCD

If the question asks for the smallest shared multiple, use LCM. If it asks for the greatest shared factor, use GCD.

Stopping At A Common Multiple That Is Not The Least One

For $6$ and $8$, both $24$ and $48$ are common multiples, but only $24$ is the least common multiple.

Using The Prime Factor Rule Without Prime Factorization

The "take the larger exponent" rule applies after the numbers are written as prime factorizations of positive integers.

A Fast Check

After finding an LCM, test two things:

1. Is your answer divisible by each original number?
2. Is there any smaller positive common multiple?

For the prime factor method, the second check is usually built into the method itself.

Try Your Own Version

Try finding the LCM of $15$ and $20$ in two ways: by listing multiples and by prime factorization. If you want a second check on larger numbers, a math solver can help verify the factorization and the final multiple.