Common Multiples

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.

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.

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:

1. The common multiples of 9 and 12 are 36 and 72 for this interval;
2. The common multiples of 9 and 15 are 45 and 90 for this interval;
3. The common multiple of 12 and 15 is 60 for this interval.

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.

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.

From the table above, observe that there are two common multiples of 11 and 13 between 130 and 300, namely 143 and 286.

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.

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:

1. The common multiples of 6 and 8 are 24, 48, 72 and 96 for this interval
2. The common multiples of 6 and 10 are 30, 60 and 90 for this interval
3. The common multiples of 8 and 10 are 4 and 80 for this interval

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.

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.

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.

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.