This chapter explores highest common factors. Before the end of this chapter you should have had a good understanding of highest common factors (HCF). The chapter explores using factor trees to find the highest common factors.
Prime numbers
Prime numbers are numbers with exactly two factors. An example of a prime number is 23. The only numbers which can be divided into 23 exactly without any reminders is 1 and 23.
Factor trees
Any number can be broken down into a product of its prime factors, for example 60 below. We can use a factor tree to help us. You can read more about factor trees here.
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We can see that;
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The numbers you see at the end of the factor tree are known as prime numbers. Here we say that they are the prime factors of 60. That is 2, 3, and 5.
Index Notation
Take for example the number 72. Using prime factors we can see that;
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The numbers 2 and 3 appear more than once, we call these numbers repeated factors. We can use index notation instead of writing out the repeated factors. This is shown below;
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The above is much shorter than before after using index notation instead.
Highest common factor (HCF)
It was necessary that you had a quick walk through of the above before we actually explored highest common factors.
Suppose we wanted to find the highest common factor of 18 and 30.
The highest common factor (HCF) is the highest number that can be divided into the given numbers without receiving reminders
18 has factors of;
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30 has factors of;
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The highest factor that appears in both lists is a 6 as shown below.
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Using factor tress to find the highest common factor (HCF)
We can use factor trees to find the highest common factor. Suppose we wanted to find the HCF of 84 and 120. The first step would be to draw the factor trees for both numbers. Let’s start with 84.
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We can see that;
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Now the 120 factor tree;
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We can see that;
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Now let’s put the finding together and pair off the factors which occur in both products.
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Next we multiply the numbers which occur in both products and the result would be the highest common factor.
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We have found that the highest common factor of 120 and 84 is 12.