Welcome to Finding the Lowest Common Multiple, the fifth and final course in this learning path! Over the previous four courses, you built a strong toolkit: factors, multiples, prime factorisation, and the Highest Common Factor. Now we will put that foundation to work in a new direction by exploring common multiples and, eventually, the Lowest Common Multiple (LCM).
In this first lesson, we will learn how to generate lists of multiples for two or more numbers, compare those lists, and spot the values that appear in every one. By the end, you will have a clear picture of what a common multiple is and a reliable method for finding one.
A Quick Refresher on Multiples
What Is a Common Multiple?
Finding Common Multiples by Listing
Working With More Than Two Numbers
Common Multiples in Everyday Life
Conclusion and Next Steps
In this lesson you learned that a common multiple of two or more numbers is any value that each of them divides into evenly. The core method is straightforward: list multiples of each number, compare the lists, and pick out the shared values. Remember that every set of whole numbers has infinitely many common multiples, and that when three or more numbers are involved, a common multiple must appear in all of the lists.
Up next, you will put these ideas into practice with hands-on tasks — generating multiples lists, spotting shared values in tables, and applying common-multiple reasoning to real-world scheduling and logistics scenarios. Once you are comfortable finding common multiples, the next lesson will build on this skill to introduce the Lowest Common Multiple, the smallest value in that shared list.
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As you may recall from earlier courses, a multiple of a whole number is the result of multiplying that number by any positive whole number. For example, the multiples of 4 are 4,8,12,16,20,… because 4×1=4, 4×2=8, and so on. Every whole number has infinitely many multiples, so the list never ends.
There is also a handy division perspective: if a number n divides evenly into a larger number m (with no remainder), then m is a multiple of n. Keep both views in mind — multiplication for generating multiples, and division for checking them.
A common multiple of two or more numbers is a value that is a multiple of each of those numbers. Put differently, every one of the given numbers divides into it evenly.
Consider the numbers 3 and 5. The number 15 is a multiple of 3 (since 3×5=15) and also a multiple of 5 (since 5×3=15). Because 15 appears in both multiples lists, it is a common multiple of 3 and 5. So is 30, and 45, and infinitely many others — there is no largest common multiple, just as there is no largest multiple of any single number.
The most direct way to find common multiples is to list several multiples of each number and then look for matches. Let us walk through an example with 4 and 6.
Step 1 — List the first several multiples of each number.
Multiples of 4
4
8
12
16
20
24
28
32
36
40
Multiples of 6
6
12
18
24
30
36
42
48
54
60
Step 2 — Compare the two rows and pick out every value that appears in both.
Scanning across, we can see that 12, 24, and 36 show up in both lists. Each of these is a common multiple of 4 and 6.
Step 3 — Verify with division. For any match you find, you can confirm it by checking divisibility. Take 24:
24÷4=6and24÷6=4
Both divisions are exact, so 24 is indeed a common multiple of 4 and 6. This quick check is especially helpful when the lists are long and you want to be sure you haven't misread a value.
The same idea extends naturally when you have three or more numbers. A common multiple must now appear in every list, not just two. Let us try 2, 3, and 5.
Multiples of 2:2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,…
Multiples of 3:3,6,9,12,15,18,21,24,27,30,…
Multiples of 5:5,10,15,20,25,30,…
The first number that sits in all three lists is 30. We can double-check:
30÷2=15,30÷3=10,30÷5=6
All exact — so 30 is a common multiple of 2, 3, and 5. Notice that you may need to write out more multiples than usual when working with three numbers, because the common values tend to be farther apart.
Common multiples pop up whenever two or more repeating cycles eventually line up. Imagine one gym class runs every 4 days and another runs every 6 days, both starting on Day 0. The days on which both classes are held are exactly the common multiples of 4 and 6: Day 12, Day 24, Day 36, and so on.
The same thinking applies to delivery schedules, rotating work shifts, or any situation where independent cycles need to coincide. Recognising that kind of problem as a common-multiple question is half the battle; the listing technique you just practised handles the rest.
