## Lowest Common Denominators

Fractions can only be added or subtracted when they have the same denominator. When the fractions do *not* have the same denominator, you must convert them into equivalent fractions with a common denominator. In most cases you will be required to find the **lowest common denominator**, since keeping the numbers as low as possible will make the problem easier.

There are a couple of ways to find the lowest common denominator for a group of fractions. The first is to list the multiples of the largest denominator until you find one that is also a multiple of the other denominators. For example, let’s consider the following group of fractions: ⅔, ¾, ⅞. Since 8 is the largest denominator, we will work through the multiples of 8 until we find one that is also evenly divisible by 2 and 3:

- 8 × 1 = 8, which is evenly divisible by 4 (8 ÷ 4 = 2) but not by 3 (8 ÷ 3 = 2 ⅔). This is not a common denominator.
- 8 × 2 = 16, which is also evenly divisible by 4 (16 ÷ 4 = 4) but not by 3 (16 ÷ 3 = 5 ⅓). This is not a common denominator.
- 8 × 3 = 24, which is evenly divisible by both 4 (24 ÷ 4 = 6) and 3 (24 ÷ 3 = 8). This
*is*the lowest common denominator!

Another way to find the lowest common denominator involves the prime factorizations of each number. The **prime factorization** of a number is the list of prime numbers that can be multiplied together to produce that number. Remember that a **prime number** can only be evenly divided by itself and 1. Here are a few examples:

- The prime factorization of 14 is 2 × 7
- The prime factorization of 18 is 2 × 3 × 3
- The prime factorization of 32 is 2 × 2 × 2 × 2 × 2
- The prime factorization of 51 is 3 × 17

To find the lowest common denominator for a group of fractions, then, first write out the prime factorization for each denominator. Then, for each different prime factor, take the largest number of times it appears in any one factorization. This is perhaps a little confusing, so let’s look at an example with the fractions 11/15 and 17/40. The prime factorizations of the denominators are as follows:

- 15: 3 × 5
- 40: 2 × 2 × 2 × 5

So, if we take each different factor the largest number of times it appears, we end up with 2, 2, 2, 3, and 5. We can now find the lowest common multiple of 15 and 40 (and, thereby, the lowest common denominator of 11/15 and 17/40) by multiplying these factors together: 2 × 2 × 2 × 3 × 5 = 120. The lowest common denominator of 11/15 and 17/40 is 120.

This second method is better for dealing with larger denominators. Let’s look at one more example of how it works. Take the fractions ¾, 1/20, and 13/14. The prime factorizations of the denominators are as follows:

- 4: 2 × 2
- 20: 2 × 2 × 5
- 14: 2 × 7

To find the lowest common denominator, then, we set up a multiplication problem that includes each different factor the maximum number of times it is used in any of the factorizations: 2 × 2 × 5 × 7 = 140, so 140 is the lowest common denominator.