LCM

Q. Find the least common multiple of the numbers 4, 6 and 8

A. We take each number in turn and write out a list of its multiples:

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28...

Multiples of 6 are: 6, 12, 18, 24, 30, 36...

Multiples of 8 are: 8, 16, 24, 32, 40, 48...

If you look at the 3 lists of numbers you will see that the lowest common number in all 3 lists is number 24, so we state that "the LCM of 4, 6 and 8 is 24"

Q. Find the lowest common multiple of 3, 5 and 6

A. As above we take each number in turn and write out a list of its multiples

Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39...

Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60...

Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66...

If you scan the list you will see that the lowest common/least common factor of 35 and 6 is 30

Sometimes the questions are given as a simple straightforward numerical problem, other times you may be given a scenario in which you have to derive what in fact the question is asking you, and then answer it. Let us take a look at an example of this:

Q. In a certain section of the local garden centre there is a number of plants between 95 and 205. Garden centre staff have noticed that these can be arranged exactly in rows of 25 or in rows of 30. With this in mind how many plants are there?

A. What we are looking for here is the lowest common multiple of the numbers 25 and 30 which sits between 95 and 205. If we write out the multiples we should be able to spot this in our lists.

Multiples of 25 are: 25, 50, 75, 100, 125, 150, 175, 200, 225...

Multiples of 30 are: 30, 60, 90, 120, 150, 180, 210....

You should be able to spot from the list that the lowest common multiple/least common multiple in these lists is 150.

Q. In your local branch of a well-known DIY shop, we have between 240 and 300 cans of paint. This can be arranged in either batches of 40 or batches of 70. How many cans of paint have we got?

A. The question is asking us to establish the least common multiple of 40 and 70 which falls between 240 and 300. We do this in exactly the same way as previously, by listing a suitable number of multiples of both numbers

Multiples of 40 are: 40, 80, 120, 160, 200, 240, 280, 320

multiples of 70 are: 70, 140, 210, 280, 350

When you examine the lists you should be able to see that the least common multiple in this case is 280.

Q. I visit my grandmother every 4 days, but my brother who lives a little bit further away can only make it every 5 days. We both visited today so how many days will it be before our visits coincide again?

A.We are looking for the least common multiple of 4 and 5, in exactly the same way as we have done previously we now list the factors

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28...

Multiples of 5 are: 5, 10, 15, 20, 25, 30...

From the lists you should be able to see that our visits will coincide again in 20 days time, as 20 is the least common multiple.

Sometimes the questions can be a little bit more complicated and can be multi-part, the secret here is to simply study the question and decipher what you are actually being asked. Usually the hardest part in questions like this is working out what the question is asking you for, once you've done that providing the answer is usually straightforward. Let us now take a look at one of these multi-part type questions.

Q. 2 cyclists, Bob and Alice are cycling around a circular course. They leave the start line at the same time and need to do 10 complete circuits of the course. It takes Bob 8 minutes to do one lap but it takes Alice 12 minutes.

(a) after how many minutes does Bob pass the start line? Write down all possible answers

(b) after how many minutes does Alice pass the start line? Write down all possible answers

(a) if Bob can complete one lap in 8 minutes, he will pass the start line after 8, 16, 24 and so on minutes, in other words multiples of 8 up to and including the 10th lap. This is 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 minutes. He passes the start line for the first time after 8 minutes.

(b) we are told that Alice can complete one lap in 12 minutes, she will pass the start line at the 12 minute point (for the first time) and then every 12 minutes for the next 9 laps, i.e. 12, 24, 36, 48, 60, 72, 84, 96, 108, 120.

(c) they start off at t = 0 and will simultaneously (that is the same time) pass the start line at a time which is the least common multiple of their individual lap times, if you look at the lists you will see that this is at 24 minutes.

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