Factors and multiples
In this section, we will be looking at factors. When working with numbers, it is important to be able to do a number of operations on them. The basic operations are – multiplication, division, subtraction and addition. In the case of the first two i.e. multiplication and division, when two numbers multiply each other, the two numbers are automatically viewed as the factors of their product. Alternatively, if a whole number say X, can be divided without leaving a remainder by another number say B, then B is said to be a factor of X. In simple terms,
A factor is a whole number which divides exactly into a whole number, leaving no remainder.
As you may have guessed, 1 is a factor of every whole number so in most cases, we ignore it.
An example or two should suffice.
When we multiply 4 and 5 together, we get 20.
4 x 5 = 20
This simply means that 4 and 5 are factors or 20.
Again, 30 divided by 3 is 10 implies that 3 is a factor of 30. Note that 10 is also a factor since
3 x 10 = 30
In many situations, we are concerned with complete lists of factors for a particular number. Again, let’s look at an example:
Say we want to find the complete list of factors of 52. Asides 1, the lowest possible factor is 2. Now 52 will divide 2 completely without leaving a remainder
52/2 = 26
As a result, we now know that 26 is also a factor of 52 because,
26 x 2 = 52
What other numbers are factors of 52, can you guess? Is 3 a factor?
52/3 = 17 r 1
Hence 3 is not a factor because it does not completely divide into 52 WITHOUT a remainder.
4 is also a factor because
52 / 4 = 13
And as before, we now see that 13 is also a factor.
The complete list of factors for 52 is thus
1, 2, 4, 13, 26 and 52
You can go ahead and try other numbers to see if this is true.
We can also pair these factors as follows:
52 = 1 x 52 = 2 x 26 = 4 x 13
The factors of 52 are all the numbers above – 1, 2, 4, 13, 26
Common Factors (CF)
Common factors are factors that are common to more than one whole number numbers. We already know the factors of 52, what are the factors of 30. Try it out, you should get
30 = 1, 2, 3, 5, 6, 10, 15, 30
Because
30 = 1 x 30 = 2 x 15 = 3 x 10 = 5 x 6
The numbers 1 and 2 are present in both the factors of 52 and factors of 30 therefore, they are known as common factors of 30 and 52.
Another example: Find the common factors of 40 and 50
First we find the factors of both numbers as before:
40 = 1 x 40 = 2 x 20 = 4 x 10 = 5 x 8
50 = 1 x 50 = 2 x 25 = 5 x 10
So the factors of 40 and 50 are
40 = 1, 2, 4, 5, 8, 10, 20
50 = 1, 2, 5, 10
CF = 1, 2, 5, 10
Note: You can find common factors for multiple numbers.
The largest of these numbers is known as the HIGHEST COMMON FACTOR (HCF). Hence, 10 is the HCF of 40 and 50
Test yourself with this: What is the HCF of 12 and 20?
Solution:
The factors of = , , , , and
The factors of = , , , , and .
Therefore the HCF of and = .
Multiples
In simple terms, the multiples of a number are the numbers it will divide into. Look at it this way, lets we have a number B, then its multiples are
B x 1
B x 2
B x 3
and so on.
As we can see, the multiples of a number include the number itself and are infinite. Let’s try an example:
Find the multiples of 3.
As before, multiples of 3 are
3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 12
and so on
Multiples of 5 are? You guessed it:
5, 10, 15, 20, 25,…
And multiples of 10 are?
10, 20, 30, 40, 50, …
Lowest Common Multiples (LCM)
You may be wondering what is the use of multiples as they go as far as infinity. Well good thing is we don’t need the highest common multiple as in the case of factors. However we do need the lowest common multiple. This is the lowest of the common multiples of two or more numbers.
From above, the multiples of 5 and 10 are
5 = 5, 10, 15, 20, 25, …
10 = 10, 20, 30, 40, 50, …
The lowest common multiple is the least number common to the multiples. In this case, that number is 10. Therefore, the lowest common multiple of 5 and 10 are 10!
One more example. What is the lowest common multiples of 6 and 10?
First we find the multiple of both numbers
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, ….
Multiples of 10 are 10, 20, 30, 40, 50, ….
The Lowest common multiple of 6 and 10 are 30!
