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■ It does not matter which pair of factors you choose to start a factor tree. The final result of prime factors will always be the same.

■ It is conventional to write the prime factors in ascending (i.e. increasing) order.

For example: 600 = 23 × 3 × 52

■ Here is another way to decompose a number, using division by prime numbers.

Sometimes it may be necessary to divide by 2, 3, 5, 7, 11, 13 or any prime number.

■ Some calculators can decompose a number. The Casio fx-82AU PLUS 11 and fx-100AU PLUS have the word FACT above the button. Press 600

Building Understanding

1 Sort the following list of numbers in two groups: composite numbers and prime numbers.

2 State the missing numbers in the empty boxes to complete the following factor trees.

3 State the numbers on the next set of branches for each of the following factor trees.

4 State the following prime factors, using powers.

Example 13 Expressing composite numbers in prime factor form

Express the number 60 in prime factor form.

A pair of factors for 60 is 5 × 12.

The 5 branch terminates since 5 is a prime factor.

A pair of factors for 12 is 3 × 4

The 3 branch terminates since 3 is a prime factor.

A pair of factors for 4 is 2 × 2

Both these branches are now terminated. Hence, the composite number, 60, can be written as a product of each terminating branch. The factors are listed in increasing order.

Now you try

Express the number 180 in prime factor form.

Exercise 2F

1 Express the following numbers in prime factor form.

2 Express the following numbers in prime factor form.

24

124

3 Express these larger numbers in prime factor form. Note that it can be helpful to start your factor tree with a power of 10 (e.g. 100) on one branch.

600

800

45 000

690 a 120

4 Match the correct composite number (a to d) to its set of prime factors (A to D).

150 b 144

180 c 2 × 3 × 52 C 22 × 32 × 5 d 24 × 32 D 2 × 3 × 2 × 5 × 2 b By considering the prime factor form, determine the highest common factor (HCF) of 144 and 96. b By considering the prime factor form, determine the highest common factor (HCF) of 25 200 and 77 000.

5 Find the smallest composite number that has the five smallest prime numbers as factors.

6 a Express 144 and 96 in prime factor form.

7 a Express 25 200 and 77 000 in prime factor form.

Reasoning

8 The numbers 15 and 21 can be written in prime factor form as 3 × 5 and 3 × 7. This means the lowest common multiple (LCM) of 15 and 21 is 3 × 5 × 7 (which is 105).

a Use the prime factor forms to find the LCM of 14 and 21.

b Use the prime factor forms to find the LCM of 15 and 33 c 60 = 22 × 3 × 5 and 210 = 2 × 3 × 5 × 7. Use this to find the LCM of 60 and 210.

9 Represent the number 24 with four different factor trees, each resulting in the same set of prime factors. Note that simply swapping the order of a pair of factors does not qualify it as a different form of the factor tree.

10 Only one of the following is the correct set of prime factors for 424.

A 22 × 32 × 5

B 2 × 32 × 52

C 53 × 8

D 23 × 53 a Justify why you can eliminate alternatives A and B straight away. b Why can option C be discarded as an option? c Show that option D is the correct answer.

11 a State the error in each of the following prime factor trees and/or prime decompositions.