Factors and multiples - 8

In the previous post we discussed about the multiples of a number. We hope that you have solved the given practice problems. Did you realize that the tables of various numbers are nothing but the first 10 multiples of the given number?

 Multiples and LCM

Example:

Table of 2 i.e. First 10 multiples of 2	Table of 7 i.e. First 10 multiples of 7	Table of 13 i.e. First 10 multiples of 13
2 x 1 = 2	7 x 1 = 7	13 x 1 = 13
2 x 2 = 4	7 x 2 = 14	13 x 2 = 26
2 x 3 = 6	7 x 3 = 21	13 x 3 = 39
2 x 4 = 8	7 x 4 = 28	13 x 4 = 52
2 x 5 = 10	7 x 5 = 35	13 x 5 = 65
2 x 6 = 12	7 x 6 = 42	13 x 6 = 78
2 x 7 = 14	7 x 7 = 49	13 x 7 = 91
2 x 8 = 16	7 x 8 = 56	13 x 8 = 104
2 x 9 = 18	7 x 9 = 63	13 x 9 = 113
2 x 10 = 20	7 x 10 = 70	13 x 10 = 130

Let’s explore the concept of multiple further.

Write down first 10 multiples of number 2 and 3.

First 10 multiples of 2	First 10 multiples of 3
2 x 1 = 2	3 x 1 = 3
2 x 2 = 4	3 x 2 = 6
2 x 3 = 6	3 x 3 = 9
2 x 4 = 8	3 x 4 = 12
2 x 5 = 10	3 x 5 = 15
2 x 6 = 12	3 x 6 = 18
2 x 7 = 14	3 x 7 = 21
2 x 8 = 16	3 x 8 = 24
2 x 9 = 18	3 x 9 = 27
2 x 10 = 20	3 x 10 = 30

From the table above we observe that within first 10 multiples of each 2 & 3, the common multiples are 6, 12 and 18. Here the smallest common multiple is 6.  

We know that a number has infinitely many multiple. Hence there can be many common multiples between any two numbers.

Let’s consider some more examples.

Multiples of 4 and 6

First 10 multiples of 4	First 10 multiples of 6
4 x 1 = 4	6 x 1 = 6
4 x 2 = 8	6 x 2 = 12
4 x 3 = 12	6 x 3 = 18
4 x 4 = 16	6 x 4 = 24
4 x 5 = 20	6 x 5 = 30
4 x 6 = 24	6 x 6 = 36
4 x 7 = 28	6 x 7 = 42
4 x 8 = 32	6 x 8 = 48
4 x 9 = 36	6 x 9 = 54
4 x 10 = 40	6 x 10 = 60

From the first 10 multiples of 4 and 6, the common multiples are 12, 24 and 36. The smallest common multiple is 12.

Multiples of 6, 9 and 12

First 10 multiples of 6	First 10 multiples of 9	First 10 multiples of 12
6 x 1 = 6	9 x 1 = 9	12 x 1 = 12
6 x 2 = 12	9 x 2 = 18	12 x 2 = 24
6 x 3 = 18	9 x 3 = 27	12 x 3 = 36
6 x 4 = 24	9 x 4 = 36	12 x 4 = 48
6 x 5 = 30	9 x 5 = 45	12 x 5 = 60
6 x 6 = 36	9 x 6 = 54	12 x 6 = 72
6 x 7 = 42	9 x 7 = 63	12 x 7 = 84
6 x 8 = 48	9 x 8 = 72	12 x 8 = 96
6 x 9 = 54	9 x 9 = 81	12 x 9 = 108
6 x 10 = 60	9 x 10 = 90	12 x 10 = 120

From the first 10 multiples of 6, 9 and 12, the common multiple among all is 36 which is the smallest common multiple as well.

The smallest common multiple of given numbers is called Least common multiple (LCM).

From the given examples:

·         LCM of 2 and 3 is 6.  

·         LCM of 4 and 6 is 12.

·         LCM of 6, 9 and 12 is 36.

Practice time:

Write first 10 multiples of each of the following and find the LCM.

         i.        3, 7

        ii.        5, 8

       iii.        12, 15

      iv.        16, 80

       v.        16, 24

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
.	.	.
.	.	.
.	.	.
12 x 10	=	120
12 x 11	=	132
12 x 12	=	144
.	.	.
.	.	.
.	.	.
12 x 30	=	360
.	.	.
.	.	.
.	.	.
12 x 45	=	540
.	.	.
.	.	.
.	.	.

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