A simple definition of multiples

A multiple of a number is a number obtained by multiplying the given number by another whole number.

Important idea: When we say a number is a multiple, it is always a multiple of some other number.

It doesn’t make sense to say 15 is a multiple without saying which number it is a multiple of. 15 is a multiple of many numbers like 3, 5 and 15 but it is not a multiple of most other numbers like 6, 7, 8, etc.

Using Multiplication to understand Multiples

The best way to understand multiples is through multiplication.

Example 1:

Look at this multiplication statement: 5 x 3 = 15

Think of this statement as a group of 5 multiplied 3 times:

A number obtained by multiplying 5 by a whole number is a multiple of 5. 

So 15 is a multiple of 5. 

No wonder the word “multiple” sounds so similar to “multiplication”!

We could have also thought of 5 x 3 = 15 as 5 groups of 3 each.

Thus, we can obtain 15 by multiplying 3 by 5. So 15 is also a multiple of 3.

3 x 5 = 15

So any such multiplication statement tells us that the number on the right (15) is a multiple of the other two numbers on the left (3 and 5).

Of course a number has many multiples. How can we find more multiples? Read on...

Using Multiplication tables to list all multiples

We can obtain many other multiples of a number using multiplication tables.

Here are the first few rows from the multiplication table of 5.

5 x 1 = 5 
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20 
… and so on.

The numbers 5, 10, 15, 20, ... are all multiples of 5.

These multiples are obtained by multiplying 5 by another whole number. On the other hand, we cannot obtain 13 by multiplying 5 by a whole number. So 13 is not a multiple of 5.

An activity to visualise multiples

You can use Cuisenaire rods (or fraction rods) to visualise multiples of a number.

Cuisenaire rods have different sizes and colours. The size determines the value of the rod.

Here are the steps to visualise the multiples of 5.

Step 1: Arrange all Cuisenaire rods one next to the other so we know which colour/length denotes which number.

Step 2: Pick the rod that denotes 5. Lay it horizontally. This shows 5 x 1 = 5. So this is the first multiple of 5.

Step 3: Pick another rod of length 5. Place it horizontally as an extension of the first rod. We now have a combined length of 10. What we have shown is 5 x 2 = 10. This is the second multiple of 5.

Step 4: Pick another rod of length 5 and stack it along the previous two rods. So we now have a total length which is 5 x 3 = 15. This is the third multiple of 5.

Step 5: You can continue this process a few more times to visualise the next few multiples of 5.

Interestingly, 5 x 0 = 0. This statement shows we can obtain zero by multiplying 5 by a whole number. So technically, zero is the first multiple of 5. But when referring to multiples commonly, we ignore 0 as a multiple. So 5 x 1 = 5 becomes the first multiple. See the reason for ignoring zero below.

You can even repeat this entire activity with a rod of another length to visualise its multiples.

You can see through this activity, that we can keep stacking new rods and obtaining new multiples. The process will never end.

Important idea: 
               There are infinite multiples of any number.               

How and why is zero a multiple of every number?

We have defined multiples of a number as a number obtained by multiplying by a whole number. This leads to an interesting result.

We could take any number and multiply it by the whole number 0. We will obtain zero.

E.g.
5 x 0 = 0     So 0 is a multiple of 5
3 x 0 = 0     So 0 is a multiple of 3
8 x 0 = 0     So 0 is a multiple of 8… and so on.

Zero is a multiple of every number because it can be obtained by multiplying any number by 0. 

This makes zero the first multiple of every number. But because it is a multiple of every number, it does not give us any new and insightful information about the number. Thus we often ignore zero and say the number multiplied by 1 is the first multiple of that number.

Technically, this would be called the first non-zero multiple. But often it is simply referred to as the first multiple.

Tips and Tricks to find Multiples

  • To find the first few multiples, simply remember the multiplication table. The values in the multiplication table are the first few multiples of the number.
  • The difference between consecutive multiples of a number is equal to that number.
    So if you’re given the 43rd multiple of a number and asked to find the 44th, simple add the number to the 43rd multiple. Similarly, subtracting the number from the 43rd multiple will give you the 42nd multiple.
    E.g. 43rd multiple of 8 is 344.
    So the 44th multiple of 8 will be 344 + 8 = 352.
    And the 42nd multiple of 8 will be 344 - 8 = 336.
  • Factors come in pairs such that when you multiply the two factors in the pair, they give you the number. This helps to make sure we’ve listed all factors
    E.g. Factors of 24 in pairs are: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Factors in a pair give 24 when multiplied.
    Use this to find factors in pairs. As you start checking from 1, every time you find a factor you’ll also find the other factor in the pair using this pairing technique.
    These pairs work for all numbers except perfect squares. Numbers like 25 will have the following pairs of factors: 1 and 25, 5 and 5.
  • Use divisibility rules to quickly find the first few factors. Even numbers will always have 2 as a factor. If the digits of the number add up to a multiple of 3, then 3 is a factor. Numbers ending with 0 or 
  • A number that has an even number of factors, it cannot be a perfect square. A number that has odd number of factors, is a perfect square.
    E.g. 16 has 5 factors (an odd number). So 16 is a perfect square (4 x 4 = 16)
    18 has 6 factors (an even number). So 18 is not a perfect square. 

Common mistakes or misconceptions related to Multiples

  • Misconception: The factors of a number are never ending (infinite).
    Misconception: The multiples of a number are limited in number (finite)

    Rote memorisation leads to confusion between properties of factors and multiples. Multiples of a number never end as you can keep adding the same number over and over. Factors are numbers that evenly divide a given number. They are finite.
  • Misconception: 1 is a multiple of every number.
    Misconception: 0 is a factor of every number

    Once again, this misconception occurs due to rote memorisation. 1 divides every number perfectly. So it is a factor of every number. On the other hand, if you take any number and multiply it by 0, you obtain 0. So zero is a multiple of every number
  • What are non-zero multiples of a number?
    Because 0 is a multiple of each and every number, it doesn’t add too much new information to the situation. Hence it is ignored as a multiple. A technically correct way of ignoring zero as a multiple is to ask for non-zero multiples.
    E.g. The first 5 multiples of 4 technically are: 0, 4, 8, 12 and 16. But if we are ignoring zero (As we often want to) we would say the first 5 non-zero multiples of 4 are 2, 8, 12, 16 and 20.
  • Misconception: Non-zero multiples exclude all multiples that have the digit zero.
    The first few non-zero multiples of 5 are 5, 10, 15 and 20. Notice that we include 10 and 20. Non-zero multiples exclude only the number 0 which is technically the very first multiple of 5. Multiples that have the digit zero, like 10 and 20, are not excluded.

Test your knowledge

  1. List the first 5 multiples of 7.
    ______________________________
     
  2. List all the factors of 30.
    ______________________________
     
  3. Which number is a factor of every number?
    ______________________________
     
  4. Which number is a multiple of every number?
    ______________________________
     
  5. The 36th multiple of 7 is 252. What will the 37th multiple of 7 be?
    ______________________________
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