Basic Mathematics for College Students

The LCM of 8 and 10 is a product of prime factors, where each factor is used the

greatest number of times it appears in any one factorization.

• We will use the factor 2 three times, because 2 appears three times in the
  factorization of 8. Circle 2 2 2, as shown on the previous page.


• We will use the factor 5 once, because it appears one time in the factorization
  of 10. Circle 5 as shown on the previous page.
  Since there are no other prime factors in either prime factorization, we have
  Use 2 three times.
  Use 5 one time.

Finding the LCD

The least common denominator (LCD) of a set of fractions is the least

common multiple (LCM) of the denominators of the fractions. Two ways to

find the LCM of the denominators are as follows:

• Write the multiples of the largest denominator in increasing order, until
  one is found that is divisible by the other denominators.


• Prime factor each denominator. The LCM is a product of prime factors,
  where each factor is used the greatest number of times it appears in any
  one factorization.

LCM (8, 10) 2  2  2  5  40

 

248 Chapter 3 Fractions and Mixed Numbers

Self Check 8

Add:

Now TryProblem 49

1

8



5

6

EXAMPLE 8

Add:

Strategy We begin by expressing each fraction as an equivalent fraction that has

the LCD for its denominator. Then we use the rule for adding fractions that have

the same denominator.

WHY To add (or subtract) fractions, the fractions must have likedenominators.

Solution

To find the LCD, we find the prime factorization of both denominators and use

each prime factor the greatestnumber of times it appears in any one factorization:

~

~

~

The LCD for and is 30.

To build and so that their denominators are 30,

multiply each by a form of 1.

Since 23 and 30 have no common factors other than 1,

this fraction is in simplest form.



23

30

Add the numerators and write the sum

 (^) over the common denominator 30.

14  9

30

Multiply the numerators. Multiply the denominators.

 (^) The denominators are now the same.

14

30



9

30

3

10

7

15

7

15



3

10



7

15



2

2



3

10



3

3

3

10

7

15

2 appears once in the factorization of 10.

3 appears once in the factorization of 15.

5 appears once in the factorizations of 15 and 10.

fLCD 2  3  5  30

15  3  5

10  2  5

7

15



3

10