Highest Common Factors (HCF)
Factors
A factor of a number is any number that will divide wholly (meaning with
a remainder of zero) into that original number.
For example, considering the number 10, the numbers 1, 2, 5 and 10 will
all divide into 10 giving a zero remainder. No other numbers will divide
into 10 giving a zero remainder. You can therefore say that the factors of
10 are 1, 2, 5 and 10 and conveniently write them in a list like this:
10:
1, 2, 5, 10
As a second example consider the number 12. The numbers 1, 2, 3, 4, 6
and 12 all divide wholly into 12 (with a zero remainder) and so you can
say that 1, 2, 3, 4, 6 and 12 are all factors of 12. You can write this in a
convenient list like this:
12:
1, 2, 3, 4, 6, 12
Highest Common Factor
of Two Numbers
The idea of factors can be extended to two numbers. In this case you
may be interested in finding the factors that are common to two
numbers. This is called finding the common factors of two numbers.
You can take this idea a little further to finding the highest common
factor (abbreviated to HCF) of two numbers, which is what the following
examples will illustrate.
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Example
Find the Highest Common Factor (HCF) of 10 and 12.
1.
Write lists for the factors of each of the two numbers below
one another:
In this case write lists for the factors of 10 and 12:
10:
12:
2.
1, 2, 5, 10
1, 2, 3, 4, 6, 12
The Highest Common Factor (HCF) is the highest factor that
is common to both lists:
In this case the Highest Common Factor is 2 since 2 is the only
number in both lists (aside from 1).
You say that the HCF of 10 and 12 is 2.
You do not have to say that 1 is a common factor of two numbers
since 1 is always a factor of any whole number.
Example
Consider the numbers 2 and 8. What is the HCF of these two numbers?
1.
Start by writing a list of the factors of each below one another:
2:
8:
2.
1, 2
1, 2, 4, 8
Aside from 1, the only number that is common to both lists is 2, so
you can say that the HCF of 2 and 8 is 2.
Example
What is the HCF of 16 and 20?
1.
Start by writing a list of the factors of both numbers:
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16:
20:
2.
1, 2, 4, 8, 16
1, 2, 4, 5, 10, 20
In this case there are two common factors in both lists: 2 and 4.
If this is the case you must always select the highest common
factor. So the HCF of 16 and 20 is 4.
Example
What is the Highest Common Factor of 5 and 9?
1.
The factors of each are:
5:
9:
2.
1, 5
1, 3, 9
Aside from 1, these two numbers have no common factor. You can
say that 5 and 9 have no common factor other than 1.
The HCF of 5 and 9 is 1.
Highest Common Factor
of Three Numbers
In addition to finding the Highest Common Factor of two numbers you
can also find the Highest Common Factor of three numbers.
This involves the same process of making lists of factors (this time for 3
numbers instead of 2 numbers) and choosing the highest common
number to all three lists.
Example
What is the Highest Common Factor of 4, 8 and 16?
1.
Write lists for the factors of each of the three numbers below
one another:
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In this case you must write lists for the factors of 4, 8 and 16:
4:
8:
16:
2.
1, 2, 4
1, 2, 4, 8
1, 2, 4, 8, 16
The Highest Common Factor (HCF) is the highest factor that
is common to all three lists:
In this case the Highest Common Factor is 4 since 4 is the highest
number in all three lists.
You can say that the HCF of 4, 8 and 16 is 4.
Example
What is the Highest Common Factor of 12, 9, 24?
1.
The factors of each are:
12:
9:
24:
2.
1, 2, 3, 4, 6, 12
1, 3, 9
1, 2, 3, 4, 6, 8, 12, 24
In this case the highest number in all three lists is 3 so the HCF of
12, 9 and 24 is 3.
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