# Lowest common denominator

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. It is the smallest positive integer that is a multiple of each denominator in the set. The term has entered popular culture with a different non-mathematical meaning which indicates the least sophisticated element in a particular situation.

## Examples

The LCD of

$\left\{ \frac{5}{12}, \frac{11}{18} \right\}$

is 36 because the least common multiple of 12 and 18 is 36. Likewise the LCD of

$\left\{ \frac{5}{6}, \frac{1}{4} \right\}$

is 12. Using the LCD (or any multiple of it, such as the product of the denominators) as a denominator enables addition, subtraction or comparison of fractions:

$\frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12};$

$\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6};$

$\frac{7}{9} < \frac{19}{24}\text{ since }\frac{56}{72} < \frac{57}{72}.$

The lowest common denominator of two vulgar fractions can be found by computing the least common multiple of their denominators.

## Middle school instruction

Some K–12 math standards such as the latest revision of the NCTM math standards and reform mathematics textbooks created since the 1990s De-emphasize or omit coverage of the LCD entirely in favor of finding any common, but not necessarily the lowest common denominator, or by using less powerful methods such as fraction strips or "benchmark" fractions. The "cross-multiply" method of comparing fractions effectively creates a common denominator by multiplying both denominators together.

Here is an algorithm that finds the lowest common denominator for two or more fractions:

Example: $\frac{2}{9} + \frac{1}{4} + \frac{1}{6}$

Start with the 3 denominators in an upside-down division box. The goal is to change all those denominators to 1's using this process:

 |_9_4_6_


Below this, re-write the division box as above. Now put your divisor in the left position. Start with 2 as the first divisor and see if it divides exactly into any of the three denominators (no remainder).

2|_9_4_6_    2 doesn't go into 9 exactly; don't change it. 2 goes into 4, leaving 2, and into 6, leaving 3.


You show this is by writing a new division box as your result:

 |_9_2_3_    Notice only those denominators that were divisible by 2 are changed (4 and 6; the 9 remains unchanged).


Are any of the remaining numbers divisible by 2? If so, repeat the process, using 2 as the divisor.

2|_9_2_3_    2 doesn't go into 9 exactly; don't change. 2 goes into 2, leaving 1. 2 doesn't go into 3 exactly.

 |_9_1_3_    This is your next result.


Since none of these numbers is divisible by 2, proceed to 3 as the next divisor, then 5, then 7, and so on through prime numbers if you need to. Remember, the goal is have all 1's in your division box.

3|_9_1_3_    3 goes into 9 exactly, leaving 3. 3 doesn't go into 1 exactly. 3 goes into 3 exactly, leaving 1.

 |_3_1_1_    This is your next result.  Notice that only one number isn't 1 and it's 3 which is divisible by 3.

3|_3_1_1_    3 goes into 3, leaving 1. 3 doesn't go into 1 exactly, so we don't change either of the ones.

 |_1_1_1_    Congratulations, you're done! Remember, the goal was to have all 1's in your division box.


Here are the steps again without the comments. To find the lowest common denominator (LCD) you multiply ALL of the divisors together. Notice that we used 2 twice and 3 twice as divisors.

 |_9_4_6_

2|_9_4_3_

2|_9_2_3_

3|_9_1_3_

3|_3_1_1_

2 × 2 × 3 × 3 = 36 = L.C.D.



So 36 is the lowest common denominator for 2/9 + 1/4 + 1/6.

## Other Uses

"Lowest common denominator" is often used as a figure of speech meaning the most basic, least sophisticated level of taste, sensibility, or opinion among a group of people. The phrase also appears in other places, particularly in engineering, referring to the minimum requirements or common attributes between systems. This can be useful for making different systems compatible, by taking the minimal amount of necessary features, allowing them to communicate or connect.