topic badge
Australia
Year 7

1.12 HCF and LCM

Worksheet
Highest common factor
1

For each of the following numbers:

i

List the factors.

ii

State the highest common factor.

a

6 and 12

b

9 and 24

c

14 and 32

d

28 and 42

2

Consider the following prime factorisations:

180 = 2 \times 2 \times 3 \times 3 \times 5

600 = 2 \times 2 \times 2 \times 3 \times 5 \times 5

Find the highest common factor of 180 and 600.

3

Consider the following prime factorisations:

224 = 2 \times 2 \times 2 \times 2 \times 2 \times 7

196 = 2 \times 2 \times 7 \times 7

Find the highest common factor of 224 and 196.

4

Consider the following prime factorisations:

4900 = 2 \times 2 \times 5 \times 5 \times 7 \times 7

1750 = 2 \times 5 \times 5 \times 5 \times 7

Find the highest common factor of 4900 and 1750.

5

Consider the following prime factorisations:

140 = 2^{2} \times 5 \times 7

24\ 500 = 2^{2} \times 5^{3} \times 7^{2}

Find the highest common factor of 140 and 24\ 500.

6

Consider the following prime factorisations:

32\ 000 = 2^{8} \times 5^{3}

800 = 2^{5} \times 5^{2}

Find the highest common factor of 32\ 000 and 800.

7

Consider the numbers 234 and 108.

a

Write the prime factorisation of 234 in expanded form.

b

Write the prime factorisation of 108 in expanded form.

c

Find the highest common factor of 234 and 108.

8

Find the highest common factor of the following groups of numbers:

a

8 and 16

b

18 and 24

c

21 and 33

d

36 and 48

e

42 and 24

f

72 and 108

g

121 and 143

h

150 and 560

i

48, 80, and 176

j

168, 312, and 120

k

168, 224, and 280

l

150, 450, and 375

9

The table below shows the prime factorisations of several numbers in index form.

a

Find the highest common factor of 2940 and 14\ 700.

b

Find the highest common factor of 17\ 640 and 308\ 700.

c

Find the highest common factor of 2940,26\ 250, and 308\ 700.

NumberPrime factorisation
29402^{2} \times 3 \times 5 \times 7^{2}
14\ 7002^{2} \times 3 \times 5^{2} \times 7^{2}
17\ 6402^{3} \times 3^{2} \times 5 \times 7^{2}
26\ 2502 \times 3 \times 5^{4} \times 7
308\ 7002^{2} \times 3^{2} \times 5^{2} \times 7^{3}
Lowest common multiple
10

For each of the following numbers:

i

List the first 8 multiples.

ii

State the lowest common multiple.

a

6 and 10

b

8 and 14

c

24 and 42

d

15 and 30

11

Consider the following prime factorisations:

54 = 2 \times 3 \times 3 \times 3

36 = 2 \times 2 \times 3 \times 3

Find the lowest common multiple of 54 and 36.

12

Consider the following prime factorisations:

2200 = 2 \times 2 \times 2 \times 5 \times 5 \times 11

2750 = 2 \times 5 \times 5 \times 5 \times 11

Find the lowest common multiple of 2200 and 2750.

13

Find the lowest common multiple of the following groups of numbers:

a

6 and 14

b

7 and 18

c

24 and 84

d

78 and 26

e

144 and 156

f

450 and 650

g

3, 8 and 18

h

9, 14 and 27

i

40, 260, and 140

j

128, 32, and 8

k

75, 125, and 225

l

16, 36, and 124

14

Consider the numbers 1575 and 1650.

a

Find the highest common factor of 1575 and 1650.

b

Find the lowest common multiple of 1575 and 1650.

Applications
15

Cicada species spend many years underground, emerging as a single brood every generation. Magicicada cassinii emerges every 17 years. They are hunted by the Latrodectus mactans spider, which has a 3 year breeding cycle.

a

How often do these species start their breeding cycle at the same time?

b

A genetic mutation changes the life cycle of the cicadas so the mutants emerge every 18years. How often will the mutant cicada species start their life cycle at the same time as the spiders?

16

Two airlines, JetFuel and LionAir, each land planes at an airport terminal. During the day a JetFuel plane lands every 18 minutes, and a LionAir plane lands every 16 minutes.

A shuttlebus takes passengers from the terminal to the city center, departing every 21 minutes.

a

How often does a JetFuel plane land and a shuttlebus depart at the same time?

b

Which of these two options happens more often in a single day?

A

JetFuel plane lands and the shuttlebus departs at the same time.

B

LionAir plane lands and the shuttlebus departs at the same time.

17

Allena has two pieces of cloth. One piece is 68 centimetres wide and the other piece is 32 centimetres wide. She wants to cut both pieces into strips, such that all of the strips have the same width.

a

How wide should she cut the strips if she wants the strips to be as large as possible?

b

How many strips will she end up with?

18

Mrs. Smith has three classes. Each class has 16, 28, and 36 students respectively. Mrs. Smith wants to divide each class into groups so that every group in every class has the same number of students and there are no students left over.

What is the maximum number of students Mrs. Smith can put into each group?

19

Jane has piece of cloth that has a size of 84 \text{ cm} by 112 \text{ cm}. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

ACMNA149

Investigate index notation and represent whole numbers as products of powers of prime numbers

What is Mathspace

About Mathspace