# At a Glance - Least Common Denominator

This process of cutting numbers down to size in order to get the pieces we want is referred to as finding the "lowest (or least) common denominator (LCD)." Yes, the "L" can stand for either "lowest" or "least," which mean the same thing. It could also stand for "littlest," we suppose, but that doesn't sound very professional.

The LCD is basically the same thing as the LCM (least common multiple) of two denominators. When comparing two fractions, the LCD is the smallest number that's a multiple of both denominators. Another way to say this is that the LCD is the smallest number divisible by both denominators. Another way to say this is...oh, you know what, you already have enough ways to say it.

To find the LCD quickly (because you never know when you'll only have 10 seconds to find one so that you can defuse a bomb in time), we use prime factorizations again.

Notice that, to find the LCD, it doesn't matter what the numerators of the fractions are. Usually, after finding the LCD, we replace both fractions with the equivalent versions whose denominator is the LCD. Having common denominators trumps having reduced fractions. If your teacher complains, you tell her we said so.

#### Example 1

 What is the LCD of   and  ?

#### Example 2

 What is the LCD of  and  ?

#### Example 3

 What is the LCD of  and ?

#### Example 4

 Find the LCD of  and .

#### Example 5

 Find the LCD of  and .

#### Example 6

 Find the LCD of  and .

#### Example 7

 Rewrite  and  as fractions with a common denominator.

#### Exercise 1

What's the LCD of   and ?

#### Exercise 2

What's the LCD of and ?

#### Exercise 3

What's the LCD of  and ?

#### Exercise 4

What's the LCD of and ?

#### Exercise 5

What's the LCD of and ?

#### Exercise 6

What's the LCD of  and 1?

#### Exercise 7

What's the LCD of and ?

#### Exercise 8

What's the LCD of the and ?

#### Exercise 9

For this pair of fractions, (a) find the LCD and (b) rewrite each fraction so that its denominator is the LCD you found in (a).

,

#### Exercise 10

For this pair of fractions, (a) find the LCD and (b) rewrite each fraction so that its denominator is the LCD you found in (a).

#### Exercise 11

For this pair of fractions, (a) find the LCD and (b) rewrite each fraction so that its denominator is the LCD you found in (a).