Factors and multiples - 7

Multiples of a number

Up till now we have discussed finding factors of a number i.e. to find building blocks that can be used to form the given number.

Similarly, the given number can be used to form bigger numbers. So, what are such bigger numbers called as? Such numbers are called as ‘Multiples’.

In this post, we are going to discuss more about multiples.

To understand what is meant by multiples, let’s take an example.

Let the given number be 12.

12 x 1	=	12
12 x 2	=	24
12 x 3	=	36
12 x 4	=	48
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12 x 10	=	120
12 x 11	=	132
12 x 12	=	144
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12 x 30	=	360
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12 x 45	=	540
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When the given number is multiplied by a whole number, the result obtained is called a multiple of the given number.

Hence the numbers obtained in the above table lists the multiples of 12. e.g. 12, 24, 36, 48, 120, 132, 144, 360, 540 and so on are the multiples of 12.

By looking at the table above, a question arises. How many multiples a number can have?

The answer is – ‘A number can have infinite number of multiples’.

To find a multiple of a number, simply multiply the given number by a whole number. Let’s take some more examples.

Write multiples of 9:

9 x 5	=	45
9 x 13	=	117
9 x 27	=	243

 Write multiples of 15:

15 x 4	=	60
15 x 11	=	165
15 x 17	=	255

 Write multiples of 21:

21 x 6	=	126
21 x 9	=	189
21 x 12	=	252

From the above examples, you can observe that the tables of various numbers that we learnt in earlier classes are examples of multiples of those numbers.

One more interesting fact of multiples is that "0 is a multiple of all the numbers". Remember that the result of any number multiplied by 0 is always 0. 

3 x 0	=	0
26 x 0	=	0
135 x 0	=	0

Practice time:

Find the 5 multiples each of the following numbers: 

i	3
ii	4
iii	8
iv	11
v	13
vi	18
vii	22
viii	27
ix	30
x	35

Factors and multiples - Answers

Factors and multiples – 6

Find the HCF:

i	HCF of 20 & 45	=	5
ii	HCF of 16 & 56	=	8
iii	HCF of 25 & 90	=	5
iv	HCF of 48 & 108	=	12
v	HCF of 42, 70	=	14
vi	HCF of 24 & 57	=	3
vii	HCF of 60 & 70	=	10
viii	HCF of 81 & 117	=	9

Factors and multiples – 5

 List the factor pairs of the given number:

 

i	54	=	1 x 54		ii	63	=	1 x 63
		=	2 x 27				=	3 x 21
		=	3 x 18				=	7 x 9
		=	6 x 9
iii	75	=	1 x 75		iv	87	=	1 x 87
		=	3 x 25				=	3 x 29
		=	5 x 15
v	99	=	1 x 99		vi	128		1 x 128
		=	3 x 33					2 x 64
		=	9 x 11					4 x 32
								8 x 16
vii	144	=	1 x 144
		=	2 x 72
		=	3 x 48
		=	4 x 36
		=	6 x 24
		=	8 x 18
		=	12 x 12

 

Factors and multiples - 6

Highest Common Factor (HCF) / Greatest Common Divisor (GCD)

In the previous post, we discussed about finding factors of a given number. Let's take our discussion further and list factors of numbers 32 and 48. 

Factors of 32

32	=	1 x 32
	=	2 x 16
	=	4 x 8

Factors of 48

48	=	1 x 48
	=	2 x 24
	=	3 x 16
	=	4 x 12
	=	6 x 8

∴ Factors of 32: 1, 2, 4, 8, 16, 32

∴ Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

∴ Factors that are common between both 32 and 48: 1, 2, 4, 8, 16

Though there are 5 common factors between 32 and 48, the highest common factor is 16. Therefore, 16 is called as Highest Common Factor (HCF)/ Greatest Common Divisor (GCD) of the numbers 32 and 48.

But where do we apply our knowledge of HCF in our daily life?  

Suppose I have 2 pieces of cloth of length 90 cm and 144 cm each. I want to cut them into strips of equal and maximum possible lengths so that there is no extra cloth. What should be the length of each strip? 

Solution:
Find the HCF of 90 and 144

Step 1:
List the factors of 90 and 144
Factors of 90  : 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144

Step 2:
List factors common between factors of 90 and 144  
Common factors: 1, 2, 3, 6, 9, 18

Step 3:
Note the highest common factor (HCF)
∴ HCF of 90 and 144: 18

∴ Each cloth should be cut into strips of length 18 cm so that strips are of equal and maximum possible length. 

Can you think of more practical applications of HCF? Do let me know through comments.

Practice time:
Find the HCF

i	20, 45	v	42, 70
ii	16, 56	vi	24, 57
iii	25, 90	vii	60, 70
iv	48, 108	viii	81, 117

Answers