Least Common Denominator (LCD) Calculator

1. What is a Least Common Denominator (LCD)?
2. How to calculate LCD?

    What is a Least Common Denominator (LCD)?

The least common denominator, also known as lowest common denominator or smallest common denominator of a set of fractions with denominators (a, b, c...) is the smallest positive integer that is divisible by each denominator in the set. In the simplest case we have just two numbers, a and b, and we can use the notation LCD(a, b). The LCD is the least common multiple (LCM) of the fractions' denominators.

The least common denominator calculator will help you find the LCD you need before adding, subtracting, or comparing fractions.

One way to understand the least common denominator is to list all whole numbers that are multiples of the two denominators. For example, for the fractions 1/3 and 2/5 the denominators are 3 and 5.

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33...
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50...

The multiples are common for both numbers are called common multiples, while the one with the lowest value is the least common multiple (LCM) and it is the least common denominator (LCD) for these fractions. In this case that is the number 15. You can verify this using our LCD calculator.

    How to calculate LCD?

The easiest way is, of course, to use our least common denominator calculator above, as it can handle LCD calculations for many denominators at once and you can enter them any way you like - separated by commas, spaces, tabs, new lines, etc. However, if you are not allowed to or don't want to use a calculator and need to do the math by yourself, you can use an algorithm or prime factorization, through which the calculations are simplified.

An algorithm that is simple to follow, but may take a large number of steps(!) is thus: given a finite sequence of positive integers X = (x₁, x₂, ..., x_n) where n is larger than 1, start the algorithm and on each step m increase the least element of the sequence by adding to it the corresponding element from the initial sequence, resulting in a new sequence in which all elements remain the same, but the lowest value has been increased. Stop when all elements in the sequence are equal, as their common value is the LCD of sequence X (LCD(X)).