[Please watch the video attached at the end of this blog for a visual explanation of factors and multiples]
Often mistaken for one another, factors and multiples are as different as they sound! 🙈
There is no need to be worried though, as the lesson is quite simple in reality, and in this blog post, I will guide you as to what factors and multiples are, as well as the most important topics that will be questioned alongside it, since this topic will appear as singular short questions, or may be part of a bigger problem with more marks.
What are Factors and Multiples in Mathematics?
Factors are numbers that divide another number given perfectly, meaning, it does not have a remainder.
Multiples on the other hand is a number that can be divided by another number a particular number of times, and this too does not have a remainder.
While this may seem confusing when reading it at first glance, it simply means that factors divide another number, and multiples are divided BY another number.
To start off things, we must learn one of the most important things in Mathematics, that is, the Order of Operations.
Order of operations
This teaches us how your mathematical operations should be carried out.
Abbreviated as BIDMAS, this stands for Brackets, Indices, Division, Multiplication, Addition, and Subtraction.
Hence, if you ever encounter a problem with either of these, follow the order from top to bottom to solve the sum correctly!
Another important part of today’s lesson is the product of prime numbers.
Product of prime numbers
Most of the numbers we have can be broken down and written as a product of prime numbers. In our article on Introduction To Number Theory, we found out that a prime number is a number that is multiplied by 1 and the number itself. (2 is a prime number because it can be multiplied by 1 and 2 itself).
Some common prime numbers are 2, 3, 5, and 7.
Ex: To find out how 36 is a product of prime numbers, there are a few easy steps to take.
If we first divide 36 by 2, the answer we get would be 18. 18 can be divided again by 2, which will give us 9 as an answer. As 9 cannot be divided by 2 completely, we will move on to the next prime number which is 3. Using three, we can divide 9, resulting in the answer 3. 3 can further be divided using 3, resulting in the answer 1.
So we can write this as 36 = 2 ✕ 2 ✕ 3 ✕ 3
And if we consider this in squared format 36 = 2² ✕ 3²
Using this method, you can find any number as the product of prime numbers!
Another common question that is asked in this lesson is what the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are.
Highest Common Factor (HCF)
The Highest Common Factor (HCF) is defined as the highest number among all the common factors of the numbers questioned.
Ex: If we wish to find out the Highest Common Factor (HCF) between 12 and 16, we can use two methods.
Writing down the factors of both numbers
First, write down the factors of both these numbers. This just means you need to write down all the possible numbers you can multiply with one another to get 12.
Factors of 12:
1 ✕ 12 = 12
2 ✕ 6 = 12
3 ✕ 4 = 12
So 1, 12, 2, 6, 3, and 4 are factors of 12.
Similarly, in finding factors of 16
1✕16 = 16
2 ✕ 8 = 16
4 ✕ 4 = 16
So 1, 2, 4, 8, and 16 are factors of 16.
When you compare the factors of both these numbers, we can see that they both have 1, 2, and 4 as common factors. However, remember, we need to find the HIGHEST common factor. Among 1,2 and 4, the highest number is 4.
Therefore, the Highest Common Factor of 12 and 16 is 4.
Prime factor method.
12 as a product of prime factors:
12 = 2 ✕ 2 ✕ 3
16 as a product of prime factors:
16 = 2 ✕ 2 ✕ 2
2 appears as a prime factor in both 12 and 16. It appears twice. Therefore, HCM can be found by multiplying the repeated prime factors together.
HCF = 2 ✕ 2 = 4
Lowest Common Multiple (LCM)
The lowest common multiple (LCM) is sometimes referred to as the least common multiple. Worry not, both of these words mean the same thing! 😄
The lowest common multiple means the smallest integer that is common to 2 or more numbers.
Finding the LCM is pretty simple as well.
Listing down multiples of given numbers
To find out the LCM of 6 and 10, the 1st thing that must be done is to list down the multiples of 6 and 10.
Multiples of 6:
6, 12, 18, 24, 30, 36… etc
Multiples of 10:
10, 20, 30, 40… etc
When looking at the numbers listed above, we can easily point out that 30 is the lowest common multiple for both of them. Therefore, the LCM of 6 and 10 is 30.
There is another method to find out the LCM, and this is referred to as the Prime factor method.
Prime Factor Method
In this method, you must break down both numbers using prime factors. If I use the same example as before,
6 = 2 ✕ 3
10 = 2 ✕ 5
We can see that 2 is found in both 6 and 10, so we take 2 separately. As 3 and 5 cannot be found under both numbers, we need to consider them individually.
LCM = 2 ✕ 3 ✕ 5
Therefore LCM = 30
You can find another example while watching the video explanation. ( A fun exercise: See if you can find the LCM before the instructor shows you how to)
Helpful points for practicing factors and multiples for your upcoming Cambridge IGCSE Mathematics exams.
Just like any other subject, Mathematics relies heavily on how well you prepare in advance for your exams. To make things easier, here are a few pointers that may help you get ready for your upcoming Cambridge IGCSE Mathematics exams.
Study the difference between multiples and factors.
Practice sample questions – Practise, practice, practice. Questions on factors and multiples can be both simple and hard, and so do both types of questions for good measure. Our question banks have questions of different difficulty levels that you can practice before your actual exams.
Practice past paper questions to familiarise yourself with the format of the questions and paper.
Ask for help – You are never too late to ask for assistance if you’re struggling. Reach out to us at Tutopiya to find the right tutor to help you pass your exams with flying colors!
Watch the video below for a visual explanation of factors and multiples and attempt the quiz to challenge yourself!