Indices and Factors
Prime Factors, HCF and LCM

Any positive whole number can be written as the product of a number of prime factors.

For example,

20 = 22 × 5
or 180 = 22 × 32 × 5

Note

A prime number is a positive whole number with exactly two factors; 1 and itself.

The first few prime numbers are 2, 3, 5, 7, 11, ...

Worked Examples

1

Write the number 276 as a product of prime numbers.

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Write 276 as a product of two factors:

276 = 2 × 138

But  138 = 2 × 69 so 276 = 2 × 2 × 69
But  69 = 3 × 23 so 276 = 2 × 2 × 3 × 23

This expression contains only prime numbers, so

276 = 22 × 3 × 23

This is called the product of prime factors.

Another important concept is that of the highest common factor (HCF) of two (or more) positive integers. The HCF is the largest number which is a factor of both (or all) the numbers.

2

Find the HCF of 120 and 105.

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Expressing both 120 and 105 in terms of their prime factors gives

120 = 2 × 2 × 2 × 3 × 5

105 = 3 × 5 × 7

It is easy now to see that the highest common factor is 3 × 5 = 15.
3
(a)

Write the numbers 660 and 470 as the product of prime factors.

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660 = 2 × 330
= 2 × 2 × 165
= 2 × 2 × 3 × 55
= 2 × 2 × 3 × 5 × 11

So as a product of prime factors,

660 = 22 × 3 × 5 × 11

470 = 2 × 235
= 2 × 5 × 47

So as a product of prime factors,

470 = 2 × 5 × 47
(b)

Find the largest common factor that will divide into both 660 and 470.

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To find the largest common factor that will divide into both 660 and 470, look at the factors common to each of the products of primes.

The numbers that appear in both are 2 and 5, so the largest number that will divide into both 660 and 470 is 2 × 5 = 10.

So 10 is the HCF of 660 and 470.

A related concept is that of the lowest common multiple, LCM, of two (or more) positive integers. This is the lowest number into which the two (or all) numbers can divide exactly.

4

Find the LCM of 24 and 60.

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One way to find the LCM is to write out multiples of each number.

For example,

multiples of 24 are 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, ...

multiples of 60 are 60, 120, 180, 240, 300, 360, ...

It is easy to see that 120 is the LCM of 24 and 60.

Another way is to express 24 and 60 in terms of their prime factors:

24 = 2 × 2 × 2 × 3 = 2 × (2 × 2 × 3)

60 = 2 × 2 × 3 × 5 = 5 × (2 × 2 × 3)

Noting that (2 × 2 × 3) is common to both numbers, the LCM is given by

5 × 24 = 120 or 2 × 60 = 120. So the LCM = 120

Note that 120 ÷ 24 = 5 and 120 ÷ 60 = 2.

Exercises

Which of the following are prime numbers?

Click on the prime numbers to underline.

1 ,   2 ,   3 ,   5 ,   7 ,   9 ,   13 ,   15 ,   18 ,   19 ,   21 ,   23 ,   25

Write each number below as a product of prime factors.

You must use indices where appropriate, e.g. 33 not 3x3x3 .
Write the prime factors in increasing order.

Type ^ sign to use index e.g.: a^b → ab
and * to use × .
(a)
10
(b)
42
(c)
68
(d)
168
(e)
250
(f)
270
(g)
429
(h)
825
(i)
1001

Find the highest common factors of each pair of numbers below.

(a)
36, 42
(b)
30, 42
(c)
45, 105
(d)
42, 50
(e)
50, 80
(f)
70, 315
(g)
216, 240
(h)
156, 234
(i)
735, 1617

Find the HCF for each set of three numbers given below.

(a)
20, 35, 105
(b)
90, 225, 405
(c)
16, 24, 56
(d)
200, 210, 220
(e)
72, 168, 312
(f)
330, 450, 630
(g)
216, 324, 432
(h)
660, 572, 528
(i)
1008, 1260, 1764

Find the LCM of

(a)
15 and 35
(b)
12 and 20
(c)
28 and 49
(d)
19 and 15
(e)
20 and 42
(f)
81 and 192

Find the LCM of each of the following sets of numbers.

(a)
8, 12, 40
(b)
36, 8, 12
(c)
25, 10,15
(d)
9, 8, 72
(e)
90, 80, 72
(f)
22, 10, 8