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