Least Common Denominator Examples

What is the LCD of   and  ?

It's 35, 'cause that's the smallest number that's divisible by both 5 and 7.

What is the LCD of  and  ?

Our LCD is 4, since 4 is the smallest number that's divisible by both denominators.

What is the LCD of  and ?

It's 12, because 12 is the smallest number divisible by both 4 and 6.

Find the LCD of  and .

First, we write out the prime factorization of each denominator. That's going to give us 3 × 7 and 2 × 3 × 5. To get our LCD, we simply multiply one of each type of prime number together. 

So we'll need to hang onto our 3 × 7, but we can eliminate the 3 from the second set of prime factors since it's a repeat. Our LCD will be 3 × 7 × 2 × 5 = 210. Not that small of a number, but it's smaller than it would be if we simply took 21 × 30.

Find the LCD of  and .

First write out the prime factorizations of the denominators:

 and 

As before, we've got exactly one repeated 3, so we can throw one of them out. Therefore, rather than taking 2 × 2 × 3 × 3 × 3 × 5, we just have to take 2 × 2 × 3 × 3 × 5. Only slightly less work, but we'll take it. It's not like we're making commission on this.

LCD = 2 × 2 × 3 × 3 × 5 = 180

Find the LCD of  and .

Write out the prime factorizations of the denominators:

 and .

In this case, the denominators have no factors in common. (Which makes it a pretty sure bet they're going to hook up.) In this case, the LCD is merely the product of the two denominators: 51 × 10 = 510. That's a pretty big denominator, but we'll have to live with it. We should at least ask it to start paying rent.

Rewrite  and  as fractions with a common denominator.

You can figure out the LCD for these fractions in one of two ways. One: you can do the legwork, or two: you can look back up the page and realize that we already tried this one. Keen observation will get you far in this world.

We already know that the LCD for these two fractions is 180. That means that, instead of and , we'll instead be working with the equivalent fractions  and . Please show them to their cubicles.