Basic Mathematics for College Students

1.8 The Least Common Multiple and the Greatest Common Factor 91

The Least Common Multiple (LCM)

The least common multiple (LCM)of two whole numbers is the smallest

whole number that is divisible by both of those numbers.

When finding the LCM of two numbers, writing both lists of multiples can be
tiresome. From the previous definition of LCM, it follows that we need only list the
multiples of the larger number. The LCM is simply the first multiple of the larger
number that is divisible by the smaller number.For example, to find the LCM of 3
and 4, we observe that

The multiples of 4 are: 4, 8, 12, 16, 20, 24,

4 is not 8 is not 12 is
divisible by 3. divisible by 3. divisible by 3.
Since 12 is the first multiple of 4 that is divisible by 3, the LCM of 3 and 4 is 12.
As expected, this is the same result that we obtained using the two-list method.

  

Recall that one

number is divisibleby

another if, when

dividing them, we get

a remainder of 0.

p

Finding the LCM by Listing the Multiples of the Largest Number

To find the least common multiple of two (or more) whole numbers:

1. Write multiples of the largest number by multiplying it by and
   so on.


2. Continue this process until you find the first multiple of the larger number
   that is divisible by each of the smaller numbers. That multiple is their
   LCM.

1, 2, 3, 4, 5,

EXAMPLE (^2) Find the LCM of 6 and 8.
StrategyWe will write the multiples of the larger number, 8, until we find one
that is divisible by the smaller number, 6.
WHYThe LCM of 6 and 8 is the smallest multiple of 8 that is divisible by 6.
Solution
The 1st multiple of 8: 8 is not divisible by 6. (When we divide, we get
a remainder of 2.) Since 8 is not divisible by
6, find the next multiple.
The 2nd multiple of 8: 16 is not divisible by 6. Find the next
multiple.
The 3rd multiple of 8: 24 is divisible by 6. This is the LCM.
The first multiple of 8 that is divisible by 6 is 24. Thus,
LCM (6, 8) 24 Read as “The least common multiple of 6 and 8 is 24.”

8  3  24 

8  2  16 

8  1  8 

Self Check 2

Find the LCM of 8 and 10.

Now TryProblem 25

We can extend this method to find the LCM of three whole numbers.

EXAMPLE (^3) Find the LCM of and 10.
StrategyWe will write the multiples of the largest number, 10, until we find one
that is divisible by both of the smaller numbers, 2 and 3.
WHYThe LCM of and 10 is the smallest multiple of 10 that is divisible by
2 and 3.

2, 3,

2, 3,

Self Check 3

Find the LCM of 3, 4, and 8.

Now TryProblem 35