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By eMATHinstruction
In the early units of our middle school courses, one goal is to find patterns that go beyond “just multiplying numbers.” Multiples are a perfect example. They show up everywhere in math, from fractions to equations to problem solving. You’ll use multiples and least common multiples (LCMs) so often in later math that it’s worth slowing down now and really understanding the why behind them now.
In our recent lesson on multiples of whole numbers, we explored what happens when you multiply a number again and again. Although the idea seems simple, it holds a lot of power once you see the structure behind it.
You can follow along with this topic in our teacher guided lesson video.
A multiple of a number is what you get when you take that number and multiply it by a whole number like 1, 2, 3, 4, and so on.
If you take the number 2 and list out its multiples, you get:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.
If you’ve ever counted by twos, you were actually listing multiples of 2. The same happens when you count by fives, tens, or any other number.
Think of multiples as “how many you have when you collect that number again and again.”
One 3 gives you 3.
Two 3s give you 6.
Ten 3s give you 30.
This view becomes especially helpful once you start comparing lists of multiples.
Multiples aren’t lists to memorize. They show up everywhere.
Imagine someone packing eggs in cartons of six. If every carton must be full, then the total number of eggs they have must be a multiple of 6. That’s why someone carrying 32 eggs can’t have all full cartons. Thirty-two doesn’t appear in the list:
6, 12, 18, 24, 30, 36, etc.
This kind of reasoning pops up with boxes of books, stacks of shipping crates, trays of muffins, or anything grouped in fixed amounts. Multiples tell you what quantities are possible and what quantities aren’t.
A common multiple is a number that appears in both lists of multiples.
Take 4 and 6 for example. Their multiples begin like this:
Multiples of 4
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, etc.
Multiples of 6
6, 12, 18, 24, 30, 36, 42, 48, etc.
If you look closely, you’ll see that 12, 24, 36, and 48 all show up in both lists. These are common multiples. They are the “overlap” between the two patterns.
Common multiples tell you something very important: a number both original numbers divide into perfectly. No leftovers. No remainders. Just clean division.
The least common multiple (LCM) is often the most important one. It’s simply the smallest number that appears in both lists.
For 4 and 6, that number is 12.
Think of the LCM as the first place where the two counting patterns “shake hands.” Once that first meeting happens, every future common multiple is just another multiple of that handshake number.
For example:
24 is 2 × 12
36 is 3 × 12
48 is 4 × 12
The LCM becomes a kind of anchor point for understanding how two numbers relate to each other. Later in math, the LCM plays a major role in adding fractions, solving equations with like terms, and working with ratios.
Let’s go back to our warehouse example.
Joe is moving boxes containing 8 books each.
Tim is moving boxes containing 10 books each.
If they finish their work having moved the same number of books, what is the smallest amount that could be?
The key is recognizing that Joe total must be a multiple of 8:
8, 16, 24, 32, 40, 48, etc.
Tim’s total must be a multiple of 10:
10, 20, 30, 40, 50, 60, etc.
The first number they have in common is 40.
So if they moved the same amount, the least number of books they could have moved is 40.
That is the LCM in action. It tells you the earliest point where two different grouping systems can match.
Multiples and LCMs might feel like simple list-making at first, but they become foundational for what comes later:
By recognizing multiples and how they line up, it strengthens number sense and prepares you for much deeper math ideas that we will explore.
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Thank you for using eMATHinstruction materials. In order to continue to provide high quality mathematics resources to you and your students we respectfully request that you do not post this or any of our files on any website. Doing so is a violation of copyright. Using these materials implies you agree to our terms and conditions and single user license agreement.
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