Math Core

Hot dogs come in packs of 10, but buns come in packs of 8. How many of each do you need to buy to end up with the same number of hot dogs and buns? To answer questions like this, you need the smallest number that is a multiple of both 10 and 8.

Multiples and common multiples

A multiple of a number is what you get when you multiply it by 1, 2, 3, and so on. The multiples of 6 are 6, 12, 18, 24, 30, 36, and they go on forever.

A common multiple of two numbers is a multiple of both. Common multiples of 6 and 8 include 24, 48 and 72.

Definition

Least common multiple (LCM)

The least common multiple of two or more whole numbers is the smallest positive number that is a multiple of all of them.

Method 1: List the multiples

List multiples of the larger number, and stop at the first one the smaller number divides evenly. This is usually faster than listing both.

Worked example: Listing multiples

Find the LCM of 6 and 8.

Multiples of 8: 8, 16, 24, …

Is 8 a multiple of 6? No.
Is 16 a multiple of 6? No.
Is 24 a multiple of 6? Yes, $6 \times 4 = 24$ .

The LCM of 6 and 8 is 24.

The number line shows the same idea. Hops of 6 and hops of 8 both start at 0, and the first place they land together is 24.

Hops of 6 land on 6, 12, 18, 24. Dots mark multiples of 8: 8, 16, 24.

Method 2: Use prime factorization

LCM from prime factorizations

Write each number as a product of primes. The LCM uses every prime that appears in either number, each one as many times as it appears the most in a single number.

Worked example: Prime factorization

Find the LCM of 12 and 18.

\begin{aligned} 12 &= 2 \times 2 \times 3 \\ 18 &= 2 \times 3 \times 3 \end{aligned}

The prime 2 appears at most twice (in 12). The prime 3 appears at most twice (in 18). So the LCM is $2 \times 2 \times 3 \times 3 = 36$ .

Check: $36 \div 12 = 3$ and $36 \div 18 = 2$ . ✓

Notice the difference from the GCF. The GCF uses only the primes the numbers share, so it is small. The LCM uses all the primes needed to build each number, so it is large.

Tip

For two numbers, $\text{GCF} \times \text{LCM}$ equals the product of the numbers. For 12 and 18: the GCF is 6 and the LCM is 36, and $6 \times 36 = 216 = 12 \times 18$ . You can use this to check your work.

LCM in word problems

Look for the LCM when two things repeat on different cycles and you want to know when they line up again, or when you need equal totals from different-sized packs.

Worked example: Hot dogs and buns

Hot dogs come in packs of 10 and buns come in packs of 8. What is the smallest number of hot dogs you can buy to have exactly one bun for each hot dog? How many packs of each is that?

The total must be a multiple of 10 (hot dogs) and of 8 (buns). Multiples of 10: 10, 20, 30, 40. The first one that 8 divides is 40.

You need 40 of each: $40 \div 10 = 4$ packs of hot dogs and $40 \div 8 = 5$ packs of buns.

Worked example: Two bus routes

Route A leaves the station every 12 minutes. Route B leaves every 20 minutes. Both buses leave together at 8:00 a.m. When is the next time they leave together?

Find the LCM of 12 and 20. Multiples of 20: 20, 40, 60. Since $12 \times 5 = 60$ , the LCM is 60.

They leave together again 60 minutes later, at 9:00 a.m.

Common mistake

Don't mix up GCF and LCM. Ask yourself what the answer should look like:

Splitting things into equal groups? The answer is at most the smaller number, so use the GCF.
Repeating or building up to a match? The answer is at least the larger number, so use the LCM.