The product of two numbers is 720 and their LCM is 90. Find their HCF.
Let's Practise - 4
1. Write the first 2 common multiples of:
a. 3 and 9
b. 4 and 6
d. 5 and 15
e. 8 and 16
c. 12 and 18
f. 20 and 30
2. Find the LCM by prime factorisation method.
a. 6 and 10
b. 15 and 45
d. 42 and 91
e. 54 and 81
g. 7, 14 and 49
h. 16,48 and 64
3. Find the LCM by division method.
a. 10 and 12
b. 6 and 9
d. 28 and 35
e. 49 and 63
c. 25 and 35
f. 4, 10 and 15
i. 40, 60 and 100
c. 15 and 20
f. 112 and 160
THEME 4: PLAYING WITH NUMBERS
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Answers
Answer:
Highest Common Factor and Lowest Common Multiple
This section introduces the idea of the highest common factor and lowest common multiple of a pair of numbers.
The highest common factor (HCF) of two whole numbers
is the largest whole number which is a factor of both.
HCF Example
Consider the numbers 12 and 15:
The factors of 12 are : 1, 2, 3, 4, 6, 12.
The factors of 15 are : 1, 3, 5, 15.
1 and 3 are the only common factors (numbers which are factors of both 12 and 15).
Therefore, the highest common factor of 12 and 15 is 3.
Example Questions
Work out the answers to these questions and click the buttons marked
Click on this button below to see the correct answer
to see whether you are correct.
(a) Find the highest common factor of 20 and 30 by following the steps below:
What are the factors of 20?
What are the factors of 30?
What is the highest common factor of 20 and 30?
(b) Find the highest common factor of 14 and 12 by following the steps below:
What are the factors of 14?
What are the factors of 12?
What is the highest common factor of 14 and 12?
The lowest common multiple (LCM) of two whole numbers
is the smallest whole number which is a multiple of both.
LCM Example
Consider the numbers 12 and 15 again:
The multiples of 12 are : 12, 24, 36, 48, 60, 72, 84, ....
The multiples of 15 are : 15, 30, 45, 60, 75, 90, ....
60 is a common multiple (a multiple of both 12 and 15), and there are no lower common multiples.
Therefore, the lowest common multiple of 12 and 15 is 60.
Example Questions
Work out the answers to these questions and click the buttons marked
Click on this button below to see the correct answer
to see whether you are correct.
(a) Find the lowest common multiple of 5 and 7 by following the steps below:
What are the first ten multiples of 5?
What are the first ten multiples of 7?
What is the lowest common multiple of 5 and 7?
(b) Find the lowest common multiple of 6 and 10 by following the steps below:
What are the first ten multiples of 6?
What are the first ten multiples of 10?
What is the lowest common multiple of 6 and 10?
Although the methods above work well for small numbers, they are
more difficult to follow with bigger numbers. Another way to find the
highest common factor and lowest common multiple of a pair of
numbers is to use the prime factorisations of the two numbers.
Finding HCF & LCM with prime factorisations
We want to find the HCF and LCM of the numbers 60 and 72.
Start by writing each number as a product of its prime factors.
60 = 2 * 2 * 3 * 572 = 2 * 2 * 2 * 3 * 3
To make the next stage easier, we need to write these so that each new prime factor begins in the same place:
60 = 2 * 2 * 3 * 5
72 = 2 * 2 * 2 * 3 * 3
All the "2"s are now above each other, as are the "3"s etc. This allows us to match up the prime factors.
The highest common factor is found by multiplying all the factors which appear in both lists:
So the HCF of 60 and 72 is 2 × 2 × 3 which is 12.
The lowest common multiple is found by multiplying all the factors which appear in either list:
So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.