B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

The Lowest Common Multiple

B2.6 A1About 40 minutes

Learning goals

I can find the lowest common multiple (LCM) of two or three whole numbers.
I can use lists of multiples or prime factorization to find the LCM.
I can tell whether a problem needs the LCM or the GCF.

Words to know

multiple
The product of a number and a whole number. Multiples of 6 are 6, 12, 18, 24...

common multiple
A number that is a multiple of two or more numbers. 24 is a common multiple of 6 and 8.

lowest common multiple (LCM)
The smallest (non-zero) common multiple. The LCM of 6 and 8 is 24.

lowest common denominator
The LCM of the denominators of two or more fractions.

Remember

In Grade 6 you added fractions like 1/4 + 1/6 by finding a common denominator, and you heard that the smallest one is called the least (or lowest) common multiple. In the last lesson you found the greatest common factor. Now you will find the lowest common multiple of two or three numbers.

Let's start with a barbecue

For a Canada Day barbecue, hot dogs come in packs of 12 and buns come in packs of 8. What is the smallest number of hot dogs you can buy so that there are exactly enough buns?

The number must be a multiple of 12 and a multiple of 8.

Multiples of 12: 12, 24, 36, 48...
Multiples of 8: 8, 16, 24, 32...

The first number on both lists is 24. Buy 24 ÷ 12 = 2 packs of hot dogs and 24 ÷ 8 = 3 packs of buns.

Hot dogs cooking on a backyard barbecue grill, with a bag of hot dog buns on the side shelf. — Hot dogs and buns often come in different-size packs. The LCM tells you the smallest number you can buy so they match exactly. AI-generated image

Multiples of 8 hop along the top and multiples of 12 along the bottom. They first land on the same number at 24, the LCM.

Method 1: List multiples

List the multiples of the largest number and stop at the first one the other number(s) divide into.

LCM of 6 and 9: multiples of 9 are 9, 18. Does 6 divide 18? Yes. LCM = 18.

Method 2: Prime factorization

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

LCM of 12 and 18:

12 = 2² × 3
18 = 2 × 3²
Most 2s: 2² (from 12). Most 3s: 3² (from 18).
LCM = 2² × 3² = 4 × 9 = 36.

	GCF of 12 and 18	LCM of 12 and 18
Prime factors	Use the shared primes: 2 × 3	Use all primes, the most of each: 2² × 3²
Answer	6	36
Size	Never bigger than the smallest number	Never smaller than the largest number
Used for	Splitting into the most equal groups; simplifying fractions	When things line up again; common denominators

Comparing the GCF and the LCM of 12 and 18.

Worked example

Example 1: Three numbers

Find the LCM of 4, 6 and 10.

4 = 2², 6 = 2 × 3, 10 = 2 × 5.
Most 2s: 2² (from 4). Most 3s: 3 (from 6). Most 5s: 5 (from 10).
LCM = 2² × 3 × 5 = 4 × 15 = 60.
Check: 60 ÷ 4 = 15, 60 ÷ 6 = 10, 60 ÷ 10 = 6. ✓

LCM(4, 6, 10) = 60

Worked example

Example 2: When will they meet again?

Three community buses leave the station together at 7:00 a.m. Route A gets back every 20 minutes, Route B every 30 minutes, and Route C every 45 minutes. When are all three next at the station together?

We need the LCM of 20, 30 and 45.
20 = 2² × 5, 30 = 2 × 3 × 5, 45 = 3² × 5.
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes.
180 minutes = 3 hours. 7:00 a.m. + 3 hours = 10:00 a.m.

All three buses are together again at 10:00 a.m.

Each dot is a bus arriving. The only times all three routes are at the station together are 7:00 and 180 minutes later, at 10:00.

GCF or LCM?

Ask yourself: "Am I splitting things up, or waiting for things to line up?"

Splitting into the most equal groups or the biggest equal pieces → GCF (answer is never bigger than the smallest number).
Finding when things happen together again or the smallest amount that works for both → LCM (answer is bigger than or equal to the numbers).

Hands-on activity

Clapping and stomping

You need: a partner, and paper and a pencil.

Count slowly from 1 to 30 together.
You clap on every multiple of 4. Your partner stomps on every multiple of 6.
Write down the numbers where you clap and stomp at the same time.
Predict the first one using the LCM before you start the next round.
Try again with 3 and 5, and then with 2, 3 and 4 (a third person snaps, or you snap too).

What to look for: 4 and 6 → together on 12 and 24 (LCM 12). 3 and 5 → 15 and 30 (LCM 15). 2, 3 and 4 → 12 and 24 (LCM 12).

Math talk: "After the first time you clap and stomp together, how often does it happen again?" (Every 12 counts: the common multiples are multiples of the LCM.)

Just for fun

In 2024, a group of cicadas that comes out of the ground every 13 years and a group that comes out every 17 years showed up in the same year. When will those two groups do it again? In LCM(13, 17) = 221 years. Talk about a rare party!

Your turn

Find the LCM of 4 and 6.

Find the LCM of 5 and 7.

Find the LCM of 8 and 12.

Find the LCM of 9 and 12.

Find the LCM of 2, 3 and 5.

Find the LCM of 6, 8 and 12.

Granola bars come in boxes of 6 and juice boxes come in packs of 10. What is the smallest number of each you can buy to have the same number of bars and juice boxes?

More practice: warm-up

List the first five multiples of 6.

Find the LCM of 3 and 4.

Find the LCM of 5 and 10.

Find the LCM of 2 and 9.

Find the LCM of 6 and 9.

More practice: on your own

Find the LCM of 12 and 18.

Find the LCM of 15 and 20.

Find the LCM of 14 and 21.

Find the LCM of 4, 5 and 6.

Mia has swimming every 6 days and goes to the library every 8 days. She did both today. In how many days will she do both on the same day again?

Challenge

Find the LCM of 18, 24 and 30.

For 12 and 18, multiply the GCF by the LCM. Then multiply 12 × 18. What do you notice?

Two lights on a winter display blink together at the start. One blinks every 8 seconds, the other every 14 seconds. How many times do they blink together in the first 5 minutes, not counting the start?

Check your understanding

Quiz

What is the LCM of 6 and 10?

What is the LCM of 3, 5 and 15?

Which problem is solved with the LCM?

If a = 2² × 3 and b = 2 × 5, what is the LCM of a and b?

For 8 and 12, which is true?

What is the LCM of 7 and 9?

For parents and teachers

Big idea: The Ontario curriculum (B2.6) asks Grade 7 students to determine the lowest common multiple for two and three whole numbers, along with the greatest common factor. The LCM is the lowest common denominator when adding and subtracting fractions (next lesson).

Common difficulties:

Always multiplying the numbers together. That gives a common multiple, but not always the lowest (6 × 8 = 48, but the LCM is 24).
Mixing up GCF and LCM. Ask: "Should the answer be smaller or bigger than my numbers?"
With prime factors, using only the shared primes (that gives the GCF).

How to help: Play the clap-and-stomp game and connect it to real schedules: bus routes, practices, or chores on different cycles. This lesson builds A1 skills: making connections and reasoning about which strategy fits the problem.