Basic Mathematics for College Students

92 Chapter 1 Whole Numbers

Solution

The 1st multiple of 10: 10 is divisible by 2, but not by 3. Find the

next multiple.

The 2nd multiple of 10: 20 is divisible by 2, but not by 3. Find the

next multiple.

The 3rd multiple of 10: 30 is divisible by 2 and by 3. It is the LCM.

The first multiple of 10 that is divisible by 2 and 3 is 30. Thus,

LCM (2, 3, 10) 30 Read as “The least common multiple of 2, 3,and 10 is 30.”

10  3  30 

10  2  20 

10  1  10 

2 Find the LCM using prime factorization.

Another method for finding the LCM of two (or more) whole numbers uses prime

factorization. This method is especially helpful when working with larger numbers.

As an example, we will find the LCM of 36 and 54. First, we find their prime

factorizations:

Factor trees (or division

ladders) can be used to

find the prime factorizations.

54  2  3  3  3

36  2  2  3  3

Caution! The LCM is not the product of the prime factorization of

36 and the prime factorization of 54. That gives an incorrect answer of 2,052.

The LCM should contain all the prime factors of 36 and all the prime factors

of 54, but the prime factors that 36 and 54 have in common are not repeated.

LCM (36, 54) 2  2  3  3  2  3  3  3 1,944

(36, 54)

54

6 9

2 3 3 3

36

4 9

2 2 3 3

The LCM of 36 and 54 must be divisible by 36 and 54. If the LCM is divisible by

36, it must have the prime factors of 36, which are If the LCM is divisible

by 54, it must have the prime factors of 54, which are The smallest

number that meets both requirements is

These are the prime factors of 36.

These are the prime factors of 54.

To find the LCM, we perform the indicated multiplication:

LCM (36, 54) 2  2  3  3  3  108

2  (^2)  (^3)  3  3
   

2  3  3 3.

2  2  3 3.

The prime factorizations of 36 and 54 contain the numbers 2 and 3.

We see that

• The greatest number of times the factor 2 appears in any one of the prime
  factorizations is twice and the LCM of 36 and 54 has 2 as a factor twice.


• The greatest number of times that 3 appears in any one of the prime
  factorizations is three times and the LCM of 36 and 54 has 3 as a factor three
  times.
  These observations suggest a procedure to use to find the LCM of two (or more)
  numbers using prime factorization.

36  2  2  3  3 54  2  3  3  3