We have so far seen how to recognize and conduct the basic operations of addition, subtraction, multiplication and division using numbers. These operations are merely the beginning. Mastery of these simply pave the way to more advanced, complicated and powerful operations along with one of the most interesting group of numbers. Factors and multiples represent an evolution in the application of basic operations. They help in carrying out more advanced calculations in an easier and more succinct manner.
Definition of Multiples
A multiple of a number is a number obtained by multiplying the given number by another whole number. i.e, Any number that features in the times table of a particular number is called its Multiple.
Important idea: When we say a number is a multiple, it is always a multiple of some other number.Example of Multiples
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.
Topics closely related to Factors, multiples and Primes
The image given below shows how Factors, Multiples and Prime Numbers are connected to other topics. To understand Factors and Multiples one should be proficient at topics like multiplication, division and divisibility tests. Must have a good number sense, a good grasp on operations with numbers.
Here is how Cuemath students visualise Multiples using Fraction rods
Experience this in a Cuemath class. Book a Cuemath demo class with a nearby centre.
Using Multiplication to understand Multiples
The best way to understand multiples is through multiplication.
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.
Mutiples of 5
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.
Activity to visualise Multiples
You can use Cuisenaire rods (or fraction rods) to visualise multiples of a number. You can also create these at home.
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.
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.
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.
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 the 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.
- List the first 5 multiples of 7.
- Which number is a multiple of every number?
- The 36th multiple of 7 is 252. What will the 37th multiple of 7 be?
Simple ways to learn Multiples
Like any other topic in arithmetic, the key to proper understanding is a strong grasp of the basics and practice. The most effective ways to ensure that your child excels with Multiples are:
Flash Cards: These are an excellent way of keeping concepts at your fingertips. Dedicate thirty minutes of your child’s study routine to make them go through the concepts of multiples. This exercise not only strengthens core concepts but also build mental math abilities.
Q1. Can prime numbers be multiples?
No, prime numbers cannot be multiples. Multiples of a number are numbers found by multiplying the original number by a whole number whereas prime numbers are the numbers with only two distinct factors. For example, let’s take \(2\) as the prime number as it is divisible by \(1\) and itself but \(2\) can have various infinite multiples when multiplied by the whole numbers.
Q2. What is the difference between multiple & factor?
A factor is a number that leaves no remainder behind after it divides the number. On the contrary, multiple is a number reached by multiplying a given number by another. While factors of a number are finite, multiples are infinite.