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The lowest common multiple
The lowest common multiple (or LCM) of two numbers is
the smallest number that is a multiple of both the numbers.
For small numbers we can find this by writing down the first
few multiples for both numbers until we find a number that is
in both lists.
For example,
Multiples of 20 are :
20,
40,
60,
80,
100,
Multiples of 25 are :
25,
50,
75,
100,
125, . . .
120, . . .
The LCM of 20 and 25 is: 100.
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LCM
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More LCM
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The lowest common multiple
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The highest common factor
The highest common factor (or HCF) of two numbers is the
highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by
writing down all their factors and finding the largest factor in
both lists.
For example,
Factors of 36 are : 1,
2,
3,
4,
6,
Factors of 45 are : 1,
3,
5,
9,
15,
9,
12,
18,
36.
45.
The HCF of 36 and 45 is: 9.
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HCF
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More HCF
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The highest common factor
We use the highest common factor when cancelling fractions.
For example, can you cancel the fraction
36
48
?
The HCF of 36 and 48 is 12.
÷12
We need to divide the numerator and the
denominator by 12.
36
The numerator cancels to: 36 ÷ 12 = 3.
48
The denominator cancels to: 48 ÷ 12 = 4.
The fraction
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36
48
cancels to
3
4
=
3
4
÷12
.
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Prime factor decomposition
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Prime factor decomposition
We can use prime factor decomposition to find the HCF
and the LCM of larger numbers.
Any number can be written as a product of prime numbers.
Can you write 60 as a product of
prime numbers?
Start with the smallest prime number
that divides into 60.
Now use the smallest prime number
that divides into 30.
2
60
2
30
3
15
5
5
1
Follow the same technique for 15.
The final stage sees us divide 5 by 5.
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60 = 2 × 2 × 3 × 5
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Prime factor decomposition
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Prime factor trees
Another method of writing a number as a product of prime
factors is by using a prime factor tree.
Can you write 294 as a product
of prime factors?
294
2
147
3
294 = 2 × 3 × 7 × 7
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49
7
7
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Prime factor trees
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Finding the HCF and LCM
Writing larger numbers as the product of prime factors can
make it easier to calculate the HCF and LCM of two numbers.
What is the HCF and the LCM of 60 and 294?
60 = 2 × 2 × 3 × 5
60
294 = 2 × 3 × 7 × 7
2
5
2
7
3
7
294
The HCF of 60 and 294 = 2 × 3 = 6.
The LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940.
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Finding the HCF and LCM
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Using factor trees to find LCM
It is also possible to calculate the LCM of two numbers using
prime factor trees.
What is the LCM of 54 and 40?
54
2
40
2
27
9
3
3
54 = 2 × 3 × 3 × 3
20
2
3
10
2
5
40 = 2 × 2 × 2 × 5
The LCM of 52 and 40 = 2 × 3 × 3 × 3 × 2 × 2 × 5 = 1080.
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Using factor trees to find HCF
Prime factor trees can also be used to calculate the HCF of
two numbers.
What is the HCF of 90 and 375?
90
2
375
3
45
15
3
3
125
5
5
90 = 2 × 3 × 3 × 5
25
5
5
375 = 3 × 5 × 5 × 5
The HCF of 90 and 375 = 3 × 5 = 15.
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Problem solving
When numbers are expressed as a product of their prime
factors, we can work out their values even if they are hidden
from us in the initial expression.
What are the values of a, b and c
in this expression?
a²b²c = 396, where a, b and c are all
prime numbers.
Peter thinks that the larger the number, the greater the
number of prime factors. Is he correct? Show your
working and include examples to support your claim.
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A snail’s pace
Two snails are being raced around a circular track. It takes
one snail 4 minutes to complete a full lap and the other snail
takes 7 minutes to complete one lap.
How long will it be before they
next meet at the starting point?
A third snail takes
3 minutes to complete a lap.
How long will it be before all three snails are at the
starting point again?
How many laps will each snail have completed?
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Rolling coins
Two coins with circumferences of
6 cm and 8 cm are rolled along a
flat surface.
What is the shortest distance the
coins can roll if they both have to
travel an exact number of turns?
Why can the coins not travel a distance of 56 cm
together in an exact number of turns?
Investigate what the shortest distance is that two
different coins can travel before completing an exact
number of turns.
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Flower power
A garden designer is planning his
rectangular flowerbeds.
The first bed will feature red and
yellow flowers and the display
must be as wide as possible.
Each row can only contain one
colour and all plants must be used.
If he has 60 red plants and 36 yellow plants how wide
will the beds be?
In another bed he has 3 different coloured flowers.
Using the same approach, how wide will the beds be if
he has 105 red, 60 yellow and 150 white plants?
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