Highest Common Factor and Lowest Common Multiple

The objectives of this section are to

The highest common factor (HCF) of two numbers is the largest number that is a factor of both.

The factors of 12 are 1,   2,   3,   4,   6,   12.

The factors of 15 are 1, 3, 5, 15.

So the HCF of 12 and 15 is 3.

The HCF is easy to find for some numbers, but for others it is more difficult. In harder cases, the best way to find the HCF is to use prime factors.

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both.

For example, 18 is the smallest number that is a multiple of both 6 and 9, so the LCM of 6 and 9 is 18.

The LCM for larger numbers can be found by using prime factorisation.

1

Find the HCF of:

(a)

20 and 30

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The factors of 20 are 1, 2, 4, 5, 10 and 20.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.

The HCF of 20 and 30 is 10.
(b)

14 and 12

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The factors of 14 are 1, 2, 7 and 14.

The factors of 12 are 1, 2, 3, 4, 6 and 12.

The HCF of 14 and 12 is 2.
2

Find the HCF of 60 and 72.

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Using factor trees:

60
302
152
53
60=2×2×3×5
=22×3×5
72
362
182
92
33
72=2×2×2×3×3
=23×33

The HCF is calculated using the prime factors that are common to both numbers. In this case, 2 appears twice in both, and 3 appears once in both.

So,

the HCF of 60 and 72=2×2×3
=12

To be in the HCF, the prime factor must be in both lists:

60=2×2 ×3 ×5
72=2×2×2×3×3
HCF=2×2×3

HCF = 12

Alternatively, using indices:

60=22×31×51
72=23×32×50

HCF = 12
3

What is the LCM of:

(a)

5 and 7

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The multiples of 5 are:

5,   10,   15,   20,   25,   30,   35,   40,   45,   ...

The multiples of 7 are:

7,   14,   21,   28,   35,   42,   49,   ...

The LCM of 5 and 7 = 35.
(b)

6 and 10

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The multiples of 6 are:

6,   12,   18,   24,   30,   36,   42,   ...

The multiples of 10 are:

10,   20,   30,   40,   50,   60,   ...

The LCM of 6 and 10 = 30.
4

Find the LCM of 60 and 72.

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From Example 2,

60 = 2 × 2 × 3 × 5 = 22 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32

The LCM includes all the factors from either number.

To be in the LCM, the prime factor can be in either list or in both lists:

60=2×2 ×3 ×5
72=2×2×2×3×3
LCM=2×2×2×3×3×5

LCM = 360

Alternatively, using indices:

60=22×31×51
72=23×32×50

LCM = 360
5

Find the HCF and LCM of 50 and 70.

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Using factor trees to find the prime factorisations:

50
252
55
50=2×5×5
=21×52
70
710
25
70=2×5×7
=21×51×71
HCF=21×51×70
=10
LCM=21×52×71
=350

(a)

List the factors of 21.

Give your answer in increasing order.

, , ,
(b)

List the factors of 35.

Give your answer in increasing order.

, , ,
(c)

What is the HCF of 21 and 35 ?

HCF =

Find the HCF of:

(a)
6 and 9
(b)
14 and 18
(c)
30 and 24
(d)
15 and 10

(a)

Use a factor tree to find the prime factorisation of 42.

42
(b)

Use a factor tree to find the prime factorisation of 90.

90
(c)

Find the HCF of 42 and 90.

=

What is the HCF of:

(a)
90 and 120
(b)
96 and 72
(c)
56 and 60
(d)
77 and 50
(e)
300 and 550
(f)
320 and 128

(a)

List the first 10 multiples of 8.

Give your answer in increasing order.

, , , , , , , , ,
(b)

List the first 10 multiples of 6.

Give your answer in increasing order.

, , , , , , , , ,
(c)

What is the LCM of 6 and 8 ?

What is the LCM of:

(a)
5 and 3
(b)
9 and 6
(c)
8 and 10
(d)
12 and 9
(e)
15 and 20
(f)
6 and 11

(a)

Use a factor tree to find the prime factorisation of 66.

66
(b)

Use a factor tree to find the prime factorisation of 40.

40
(c)

Find the LCM of 40 and 66.

Find the LCM of:

(a)
28 and 30
(b)
16 and 24
(c)
20 and 25
(d)
60 and 50
(e)
12 and 18
(f)
21 and 35

Two lighthouses can be seen from the top of a hill. The first flashes once every 8 seconds, and the other flashes once every 15 seconds. If they flash simultaneously, how long is it until they flash again at the same time?

sec

At a go-kart race track, Vic completes a lap in 40 seconds; Paul completes a lap in 30 seconds, and Mark completes a lap in 50 seconds. If all three start a lap at the same time, how long is it before

(a)

Paul overtakes Vic,

sec
(b)

Vic overtakes Mark?

sec