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 Factors
Factors of a given number are numbers that can perfectly divide that given number. i.e., Any number that divides another number leaving a remainder of zero is called its Factor.
The word “factors” means part of. So, think of factors as a number part that when multiplied by a whole number gives you the original number.
Example of Factors
E.g. 3 is a factor of 12 because when 3 is multiplied by a whole number it gives 12.
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 Factors using Fraction rods
Experience this in a Cuemath class. Book a Cuemath demo class with a nearby centre.
Understanding Factors using Multiplication
Just like multiples, factors can also be understood using multiplication statements.
Example of Factors
Let’s look at another example: 3 x 4 = 12
This tells us that we can obtain 12 by multiplying 3 by 4. So 12 is a multiple of 3 as well as 4.
At the same time, it tells us that 12 can be grouped or arranged in sets of 3 or 4.
If a given number can be grouped in sets of another number, then that number is a factor of the given number.
So 3 and 4 are factors of 12.
Factors and multiples are connected by the same multiplication statement. But this doesn’t give us all the factors of 12. Let’s see how to find all factors of a number.
Understanding Factors using Division
Just as we used multiplication to understand multiples, we will use division to understand factors.
Factors of 12:
Let’s think of a number, say 12. Some numbers divide 12 perfectly, while other numbers do not. Dividing a number perfectly means not having any remainder. That is, the remainder should be zero.
E.g. 12 ÷ 6 = 2, with zero remainder. We say 6 divides 12 perfectly.
But, 12 ÷ 5 will give quotient 2 with remainder 2. We say 5 does not divide 12 perfectly.
Which numbers can perfectly divide 12? We can check number one by one…
|Finding all the factors of 12|
we are checking
|Dividing 12 by that
|Does it divide
|1||12 ÷ 1 = 12, remainder 0||Yes, so 1 is a factor of 12|
|2||12 ÷ 2 = 6, remainder 0||Yes, so 2 is a factor of 12|
|3||12 ÷ 3 = 4, remainder 0||Yes, so 3 is a factor of 12|
|4||12 ÷ 4 = 3, remainder 0||Yes, so 4 is a factor of 12|
|5||12 ÷ 5 give quotient 2 and remainder 2||No|
|6||12 ÷ 6 = 2, remainder 0||Yes, so 6 is a factor of 12|
|7||12 ÷ 7 give quotient 1 and remainder 5||No|
|8||12 ÷ 8 give quotient 1 and remainder 4||No|
|9||12 ÷ 9 give quotient 1 and remainder 3||No|
|10||12 ÷ 10 give quotient 1 and remainder 2||No|
|11||12 ÷ 11 give quotient 1 and remainder 1||No|
|12||12 ÷ 12 = 1, remainder 0||Yes, so 12 is a factor of 12|
So the factors of 12 are: 1, 2, 3, 4, 6 and 12
Factors of 8:
Let’s consider another example. Can you find the factors of 8?
- Check which of the numbers 1, 2, 3, 4, 5, 6, 7 and 8 divide 6 perfectly.
- The numbers that give zero remainder are: 1, 2, 4 and 8
- So factors of 8 are 1, 2, 4 and 8.
Pro tip: Can you think of why we stopped checking at 8?
Only a number less than or equal to a given number can perfectly divide that number. Hence we stopped checking at 8 when finding factors of 8; and stopped at 12 when finding factors of 12 earlier.
The word “factors” means part of. So think of factors as a number part that when multiplied by a whole number gives you the original number. E.g. 3 is a factor of 12 because when 3 is multiplied by a whole number it gives 12.
Activity to visualise Factors
You can use Cuisenaire rods (or fraction rods) to visualise factors of a number. You can also create these rods at home.
Cuisenaire rods have different sizes and colours. The size determines the value of the rod.
Here are the steps to visualise the factors of 6.
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 6. Lay it horizontally.
Step 3: Now we will see if 6 can be divided perfectly by other numbers. We start with 1. So pick up a few rods of size one. Stack them below the 6 rod. Do a few rods of 1 stack up to create the length 6? Yes, they do. So 1 is a factor of 6.
Step 4: Pick up a few rods of size 2. Do a few rods of 2 stack up to create the length 6? Yes, they do. So 2 is a factor of 6.
Step 5: Pick up a few rods of size 3. Do a few rods of 3 stack up to create the length 6? Yes, they do. So 3 is a factor of 6.
Step 7: Pick up a few rods of size 4. Do a few rods of 4 stack up to create the length 6? No. They either fall short or they exceed the length 6. So 4 is not a factor of 6.
Step 8: Pick up a few rods of size 5. Do a few rods of 5 stack up to create the length 6? No. They either fall short or they exceed the length 6. So 5 is not a factor of 6.
Step 9: Pick up a few rods of size 6. Does the rods of size 6 stack up to create the length 6? Yes, just one rod of size 6 is enough. So 6 is a factor of 6.
Tips and Tricks to find Factors
- 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, simply 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 Factors and 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 all the factors of 30.
- Which number is a factor of every number?
Simple ways to learn Factors
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 Factors 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 factors. This exercise not only strengthens core concepts but also build mental math abilities.
Q1. What is the difference between prime factors & prime numbers?
Prime factors are any of the prime numbers that can be multiplied to give the number. Example: The prime factors of \(15\) are \(3\) and \(5\) (because \(3 \times 5 = 15,\) and \(3\) and \(5\) are prime numbers) whereas Prime numbers are numbers that have exactly two factors \(1\) and itself.
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.