Highest Common Factor
Other Names
The "Highest Common Factor" is often abbreviated to "HCF", and is also known
as:
•
•
the "Greatest Common Divisor (GCD)", or
the "Greatest Common Factor (GCF)"
So ... what is a "Factor" ?
Factors are the numbers you multiply together to get another number:
A number can have many factors:
Factors of 12 are 1, 2, 3, 4, 6 and 12 ...
... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.
(Read how to find All the Factors of a Number. In our case we don't need the
negative ones.)
What is a "Common Factor" ?
Let us say you have worked out the factors of two numbers:
Then the common factors are those that are found in both lists:
•
•
Notice that 1, 2, 3 and 6 appear in both lists?
So, the common factors of 12 and 30 are: 1, 2, 3 and 6
It is a common factor when it is a factor of two or more numbers.
(It is then "common to" those numbers.)
Here is another example with three numbers:
What is the "Highest/Greatest Common Factor" ?
It is simply the largest of the common factors.
In our previous example, the largest of the common factors is 15, so the Highest
Common Factor of 15, 30 and 105 is 15
The "Highest Common Factor" is the largest of the common factors (of two or
more numbers)
Why is this Useful?
One of the most useful things is when we want to simplify a fraction:
Example: How could we simplify
!"
!"
?
Earlier we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and
so the Highest Common Factor is 6.
So the largest number we can divide both 12 and 30 evenly by is 6, like this:
÷6
12
30
2
=
5
÷6
The Highest Common Factor of 12 and 30 is 6.
!"
!
And so
can be simplified to
!"
!
Finding the Highest Common Factor
Here are three ways:
1. You can:
•
•
•
find all factors of both numbers (go to
www.avilamaths.com/secret.php to help you),
then select the ones that are common to both, and
then choose the greatest/highest.
Example:
Two Numbers
9 and 12
Factors
9: 1,3,9 12: 1,2,3,4,6,12
Common Factors
Highest Common Factor
1,3
3
Example Simplified Fraction
9
/12 = 3/4
And another example:
Two Numbers
6 and 18
Factors
Common Factors
Highest Common Factor
1,2,3,6
6
6: 1,2,3,6 18: 1,2,3,6,9,18
Example Simplified Fraction
6
/18 = 1/3
2. You can find the prime factors and combine the common ones together:
Two Numbers
24 and 108
Thinking ...
Highest Example Simplified Common Factor
Fraction
2 × 2 × 2 × 3 = 24, and 2 × 2 × 3 = 12
2 × 2 × 3 × 3 × 3 = 108
24
/108 = 2/9
3. Sometimes you just know by thinking about it (if you know your tables really
well)
Two Numbers
Thinking ...
9 and 12
3 × 3 = 9 and 3 × 4 = 12
Highest Example Simplified Common Factor
Fraction
3
9
/12 = 3/4
But in that case you had better be careful you have found the greatest common
factor.
There is a 4th way, called the Euclidean Algorithm, which is extension. It will
guarantee you find the HCF for any 2 numbers.
** Extra: The Euclidean Algorithm (Finds HCF for 2 numbers)
1. Each line in the algorithm will have 3 variables called a, b, r.
2. To start with, a = larger of the 2 numbers and b = smaller of the 2
numbers. r = remainder when you do ‘a’ divided by ‘b’. Find r.
3. If r is not 0, reset the variables. a will become what b used to be. B will
become what r used to be. (old a disappears). Then recalculate r by
finding the remainder when new a is divided by new b.
4. Repeat step 3 until r DOES EQUAL 0. As soon as r = 0, stop, and the last
value of b is the highest common factor.
Example:
Find the HCF for 541 and 760.
a = 759. b = 540. b goes into ‘a’ once, remainder 219.
r = 219. Keep going.
a = 540. b=219. 219x2=438. So ‘b’ goes into ‘a’ twice, remainder 102.
r=102. Keep going.
a = 219. b = 102. 102*2=204. So ‘b’ goes into ‘a’ twice, remainder 15.
r = 15. Keep going.
a = 102. b = 15. 15*6=90. So ‘b’ goes into ‘a’ 6 times, remainder 12.
r = 12. Keep going.
a = 15. b = 12. ‘b’ goes into ‘a’ once, remainder 3.
r = 3. Keep going.
a = 12. b = 3. 3 goes into 12. r = 0. Last value of b is 3. So HCF must be
3.
You may think this long, but get a computer to do this and you can find
the HCF of 2 very large numbers in seconds.Below is Scratch Code. Make
this program. Try and see how it is exactly what we were doing above.
Then watch how quickly it finds the HCF of two 10 digit numbers you type
in. See how quickly it can do 20 digit numbers… 30 digit numbers… see if
you can BREAK it!!!