# Least common denominator

The least common denominator (LCD) for two or more fractions is the least common multiple (LCM) of their denominators. In order to add or subtract fractions, they must have a common denominator. While the simplest way to find a common denominator is to multiply all of the denominators of the fractions being added or subtracted, doing this will likely result in an equivalent fraction that is not simplified. Finding the LCD before performing the addition or subtraction will more likely result in the fraction is already in simplified form.

## Finding the LCD

The LCD of a set of fractions is the LCM of the denominators of said fractions. There are several ways to find the LCM. One way is by listing multiples.

### Listing multiples

Listing multiples is a fairly straightforward way to find the LCM. Simply list multiples of each of the denominators, and find the first multiple that all the denominators share.

Examples

1. What is the least common denominator of and ?

Multiples of 4: 4, 8, 12,...

Multiples of 6: 6, 12,...

LCM(4, 6) = 12, so the LCD is 12.

2. What is the least common denominator of , , and ?

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...

Multiples of 4: 4, 8, 12, 16, 20,...

Multiples of 5: 5, 10, 15, 20,...

LCM(2, 4, 5) = 20, so the LCD is 20.

We can test this by adding the fractions in example 1: If, instead of finding the LCD, we multiplied 4 and 6 to get a common denominator: We would then have to divide both the numerator and denominator by 2 to simplify the fraction.

### Prime factorization

Another way to find the LCM is through using prime factorization. To find the LCM using prime factorization, first find the prime factorization of each denominator. Given the denominators of two fractions are 16 and 40, find their prime factorizations.

Prime factorization of 16: 16 = 2 × 2 × 2 × 2

Prime factorization of 40: 40 = 2 × 2 × 2 × 5

After finding the prime factorizations of the denominators, multiply their prime factors. Pay attention to matching factors (listing them in increasing order helps) so as to only include matching factors once. For example, in 16, 2 appears as a prime factor 4 times; in 40, 2 appears 3 times. Thus, 16 and 40 share 3 factors of 2. Therefore, to find the LCM, multiply the 3 shared 2s by the other factors that they do not share:

2 × 2 × 2 × 2 × 5 = 80

Therefore, LCM(16, 40) = 80.

Example

Add and :  cannot be reduced, which confirms that 80 is the LCD of the given fractions.