What do the numbers 20 and 50 have in common? Well, both these numbers are divisible by 2, 5 and 10. We say this because no remainder will exist when we divide them by these three said numbers. This means that 20 and 50 are multiples of 2, 5 and 10 since
\[\mathbf{2}\times 10=\mathbf{5}\times 4=\mathbf{10}\times 2=20\]
\[\text{and}\]
\[\mathbf{2}\times 25=\mathbf{5}\times 10=\mathbf{10}\times 5=50\]
Looking at our derivation above, we can further infer that the numbers 2, 5 and 10 share two multiples, namely 20 and 50. These shared numbers are called common multiples. This article will demonstrate a method we can use to identify common multiples for a given set of numbers.
Recap: Multiples
To ease ourselves into this topic, let us go through a quick overview of our previous topic on multiples.
A multiple of a non-zero integer \(A\) is a non-zero integer \(C\) that can be obtained by multiplying it with another number, say \(B\).
In other words, \(C\) is a multiple of \(A\) if \(C\) is in the multiplication table of \(A\).
The multiple of a number, say \(a\), is given by the general formula,
\[\text{Multiple of}\ a=a\times z\]
where \(z\in\mathbb{Z}\). In other words,
if \(A\times B=C\) then \(A\) and \(B\) are divisors (or factors) of \(C\),
or \(C\) is a multiple of \(A\) (and also \(B\)).
To find a particular set of multiples for a given number, we can use the multiplication table.
As with our example above, the numbers 20 and 50 are multiples of 2, 5 and 10. The following table shows other multiples of 2, 5 and 10.
Number | First 6 non-zero multiples |
2 | 2, 4, 6, 8, 10, 12 |
5 | 5, 10, 15, 20, 25, 30 |
20 | 20, 40, 60, 80, 100, 120 |
A more in-depth explanation of multiples can be found in the topic called Multiples.
Definition of a Common Multiple and Method
Let us now define a common multiple.
A common multiple is a multiple that is shared between two (or more) numbers.
Identifying a common multiple(s) for a given set of numbers is fairly straightforward. Given a set of numbers, you would simply execute two steps:
Step 1: List the multiples of each number given in the set;
Step 2: Pick out any identical multiples shared from the lists written in Step 1.
Recall that there are an infinite number of multiples for any integer. With this property in mind, a restriction may be introduced in Step 1. In most cases, the question will define an interval for which the common multiples are satisfied for a given set of numbers.
For example, you may get questions that use the phrase "find the first two common multiples of 2 and 3" or "list the common multiples of 2 and 3 between 1 and 10". However, an interval restriction is not necessary here. But, it is safe to say that no individual can list all the common multiples for a given set of numbers by hand. That would take yards of ink and paper!
Important note: Although zero is indeed a common multiple for any set of numbers, you would typically list down the non-zero common multiples only (we shall look into this in the next section).
Here is an example for finding common multiples for a given set of numbers.
List all the (non-zero) common multiples of 9, 12 and 15 between 1 and 100.
Solution
The interval restriction here is that we need to list the multiples of 9, 12 and 15 between 1 and 100. We shall begin by listing these multiples using the table below.
Number | Multiples between 1 and 100. |
9 | 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 |
12 | 12, 24, 36, 48, 60, 72, 84, 96 |
15 | 15, 30, 45, 60, 75, 90 |
Looking at the table above, there are no visible common multiples of 9, 12 and 15 for this interval. However, you can deduce the following ideas:
- The common multiples of 9 and 12 are 36 and 72 for this interval;
- The common multiples of 9 and 15 are 45 and 90 for this interval;
- The common multiple of 12 and 15 is 60 for this interval.
Here is another worked example.
List the first 2 non-zero common multiples of 5 and 17.
Solution
The interval restriction here is that we need to list the first 2 non-zero multiples of 5 and 17.
Sometimes, listing multiples can be rather cumbersome, especially when the numbers are very far apart from each other. As with our case here, the difference between 5 and 17 is quite large, so listing the multiples of 5 may take a while until we can find one that is also a multiple of 17.
For situations like this, it is advised to list down the multiples of the larger number and test whether these multiples are also multiples of the smaller number. We do this by verifying that it is divisible by each other (this will be further explained in the next section).
For this example, let us write down the first few non-zero multiples of 17.
Multiples of 17: 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187,...
From this list, we can observe that 85 and 170 are indeed divisible by 5 since \(5\times 17=85\) and \(5\times 34=170\). Thus, the first 2 non-zero common multiples of 5 and 17 are 85 and 170.
Let us look at one more example before moving on to the next section.
List all the common multiples of 11 and 13 between 130 and 300.
Solution
The interval restriction here is that we need to list the multiples of 11 and 13 between 130 and 300. As before, we will start by listing these multiples using the table below.
Number | Multiples between 130 and 300 |
11 | 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297 |
13 | 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299 |
From the table above, observe that there are two common multiples of 11 and 13 between 130 and 300, namely 143 and 286.
The concept of a common multiple is primarily used to find the lowest common multiple (or LCM) between a given set of numbers. This is the smallest common multiple shared between two (or more) numbers. You can find a thorough discussion on this topic in the article: Lowest Common Multiple.
Try it yourself: Answer the following questions.
- What are the first two non-zero common multiples of 16 and 27?
- What are the common multiples of 9 and 12 between 22 and 140?
Solutions
Question 1: 432, 864
Question 2: 36, 72, 108
Properties of Common Multiples
Before we move on to more examples involving common multiples, let us establish some important properties of common multiples.
Property | Example |
A set of numbers can have more than one common multiple. | 6 is a common multiple of 3 and 6. However, this is not the only common multiple of 3 and 6. The numbers 12, 18 and 24 are also some other common multiples of 3 and 6. |
A set of numbers can have an infinite number of common multiples. | Common multiples of 5 and 8 include 40, 80, 120, 160, ... The values will keep increasing and the list will go on forever. |
The common multiple of a set of numbers is always greater than or equal to each of the numbers themselves (excluding 0). | Common multiples of 2 and 4 between 1 and 10 are 4 and 8. Notice that the multiple 4 is greater than the given number 2 but equal to the given number 4. The multiple 8 however is greater than both the given numbers 2 and 4. |
The given set of numbers divides the common multiple without leaving a remainder. These numbers are called the factors. | A common multiple of 8 and 17 is 136. Dividing 136 by each of these given numbers will not produce a remainder since \(8\times 17=136\) and \(17\times 8=136\). |
Every non-zero integer is a multiple of 0 since any non-zero integer multiplied by 0 equal 0. In most cases, we will only consider non-zero common multiples. | Since \(7\times 0=0\) and \(9\times 0=0\) then 0 is a common multiple of 7 and 9. |
Examples of Common Multiples
We shall end this topic by looking at a few more worked examples concerning common multiples.
List all the common multiples of 6, 8 and 10 between 1 and 100.
Solution
To begin, we shall list the multiples of each given number between 1 and 100. This is shown in the table below.
Number | Multiples between 1 and 100 |
6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96 |
8 | 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 |
10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 |
From the table above, we find that there are no common multiples of 6, 8 and 10 between 1 and 100. However, we can indeed conclude the following relationships:
- The common multiples of 6 and 8 are 24, 48, 72 and 96 for this interval
- The common multiples of 6 and 10 are 30, 60 and 90 for this interval
- The common multiples of 8 and 10 are 4 and 80 for this interval
Let us now move on to another example.
List the first 4 non-zero common multiples of 2, 7 and 14.
Solution
First, note that 2 is relatively located far from 7 and 14 on the number line. Thus, it is sensible to list the multiples of 7 and 14 and compare their common multiples. From here, you will check whether these common multiples are also divisible by 2.
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70,...
Multiples of 14: 14, 28, 42, 56, 70,...
Notice that the first 4 non-zero common multiples of 7 and 14 are 14, 28, 42 and 56. All four of these numbers are even which means that they are also divisible by 2. Thus, the first 4 non-zero common multiples of 2, 7 and 14 are 14, 28, 42 and 56.
We shall look at one last example involving common multiples.
List the first 3 non-zero common multiples of 3 and 19.
Solution
The difference between 3 and 19 is rather significant. So, as with our previous example, we will only list the multiples of 19 and verify whether they are also divisible by 3.
Multiples of 19: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190,...
From this list, we find that the numbers 57, 114 and 171 are also divisible by 3 since \(3\times 19=57\), \(3\times 138=114\) and \(3\times 57=171\). Therefore, the first 3 non-zero common multiples of 3 and 19 are 57, 114 and 171.
Real-world Examples Involving Common Multiples
Here is an interesting question: can we apply common multiples in real-life situations? As a matter of fact, we can! In this section, we shall demonstrate two examples of real-world scenarios that encapsulate all that we have learnt from this discussion.
Polly and Hannah decide to take turns visiting their friend, Ben, at the hospital. Polly suggests visiting Ben every 3 days while Hannah visits him every 5 days. If both of them visited Ben today, how long will it be until the next time they see him the same day again?
Solution
Here, we simply need to find the first non-zero common multiple of days 3 and 5. We shall take today as the first multiple of days 3 and 5 which is day 0. Remember, every non-zero integer is a multiple of 0 since any non-zero integer multiples by 0 equal 0 (property 5 of common multiples).
Let us now write down the common multiples of 3 and 5:
Multiples of 3: 2, 6, 9, 12, 15, 18,...
Multiples of 5: 5, 10, 15, 20,...
From both these lists, we see that 15 is the first non-zero common multiple of 3 and 5. Hence, the next time both Polly and Hannah will visit Ben together is on day 15.
Here is the final real-world example to tie this article up.
Rory and Tana are jogging around a circular running track. Rory takes 12 minutes to complete a lap while Tana takes 16 minutes. If both of them leave the starting point at the same time, write down the next two times both of them will pass the starting point together again.
Solution
Using a similar approach as the previous example, we need to locate the first two non-zero common multiples of minutes 12 and 16. Taking the first time they leave the starting point says minute 0, we can now list the multiples of 12 and 16.
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108,...
Multiples of 16: 16, 32, 48, 64, 80, 96, 112,...
Looking at the lists above, notice that 48 and 96 are the first two non-zero common multiples of 12 and 16. Thus, Rory and Tana will pass the starting together again at minutes 48 and 96.
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