Basic Mathematics for College Students

90 Chapter 1 Whole Numbers

EXAMPLE (^1) Find the first eight multiples of 6.
StrategyWe will multiply 6 by and 8.
WHYThe multiples of a numberare the products of that number and
and so on.
Solution
To find the multiples, we proceed as follows:
This is the first multiple of 6.
This is the eighth multiple of 6.
The first eight multiples of 6 are 6, 12, 18, 24, 30, 36, 42,and 48.

6  8  48

6  7  42

6  6  36

6  5  30

6  4  24

6  3  18

6  2  12

6  1  6

1, 2, 3, 4, 5,

1, 2, 3, 4, 5, 6, 7,

Self Check 1

Find the first eight multiples of 9.

Now TryProblems 17 and 85

The first eight multiples of 3 and the first eight multiples of 4 are shown below.

The numbers highlighted in red are common multiplesof 3 and 4.

3  8  24

3  7  21

3  6  18

3  5  15

3  4  12

3  3  9

3  2  6

3  1  3

4  8  32

4  7  28

4  6  24

4  5  20

4  4  16

4  3  12

4  2  8

4  1  4

If we extend each list, it soon becomes apparent that 3 and 4 have infinitely many

common multiples.

The common multiples of 3 and 4 are:

Because 12 is the smallest number that is a multiple of both 3 and 4, it is called

the least common multiple (LCM)of 3 and 4. We can write this in compact form as:

LCM (3, 4) 12 Read as “The least common multiple of 3 and 4 is 12.”

12 , 24 , 36 , 48 , 60 , 72 , p

The Least Common Multiple (LCM)

The least common multipleof two whole numbers is the smallest common

multiple of the numbers.

We have seen that the LCM of 3 and 4 is 12. It is important to note that 12 is

divisible by both 3 and 4.

and

This observation illustrates an important relationship between divisibility and

the least common multiple.

12

4

 3

12

3

 4