More about this LCD calculator

This calculator allows you to compute the LCD for a list of numbers you provide. You need to provide at least two integer numbers, and this calculator will compute the least common denominator for them. This is useful in case you are simplifying fractions, for which you need a common denominator.

Once a valid list of integers is provided, then you need to click on the "Calculate" button to get the results shown to you, with a step-by-step procedure.

Fractions calculations will be among the first things you will use a least common denominator, so that you can operate them with ease. Normally, students will simply want to multiply the denominators to get a common one, and though multiplying leads to a common denominator, it is often times not the least common denominator.

How to find the LCD?

One thing that may throw people off is that there is no "formula" to compute the LCD of a list of numbers, and you rather need to follow a procedure in order to do the calculation.

Though the calculation does not seem to create any difficulties when you are looking for the LCD of things like 4 and 6, which is easily found to be 12, things become less obvious if you have more than 2 numbers that are not so simple, like getting the LCD of 37, 63 and 85.

Steps for finding the LCD

• Step 1: Identify the list of numbers for which you are looking for the LCD and make sure they are integer. If you don't have integers, then you cannot carry on
• Step 2: Find the prime decomposition of each of the numbers in the list (this can be laborious for a long list of big numbers)
• Step 3: Collect the list of all prime numbers that appear in at least one of the numbers, and find the maximum exponents from all the appearances a prime gets
• Step 4: The least common denominator LCM is found using the formula \(LCM = \frac{a_1 \cdot a_2 \cdot a_3 \cdots a_n}{GCD}\)

For example, if we were to find the LCD of 4 and 6. The prime decompositions are

\[4 = 2^2\] \[6 = 2 \cdot 3\]

So overall list of different primes is 2 and 3. The maximum exponent found for 2 is 2, and the maximum exponent for 3 is 1. So then

\[LCD = 2^2 \cdot 3^1 = 12\]

The way this procedure works ensures that you find the lowest common denominator of the provided list of numbers.

Alternative way of finding the LCD

• Step 1: Identify the list of numbers for which you are looking for the LCD and make sure they are integer.
• Step 2: Find the prime decomposition of each of the numbers in the list
• Step 3: Collect the list of all prime numbers that appear in ALL of the numbers, and find the minimum exponents from all the appearances a common prime gets
• Step 4: Find the greatest common divisor by multiplying the common primes by their minimum exponent
• Step 5: The least common denominator LCM is found using the formula \(LCM = \frac{a_1 \cdot a_2 \cdot a_3 \cdots a_n}{GCD}\) The least common denominator is equal to the product of all primes found, raised to the maximum power of the exponents found for it

How to use the least common denominator

The least common denominator is used so to amplify a list of fractions so that all of them have the same denominator, a process that is absolutely necessary if you are dealing with sums or subtractions of fractions.

Having this common denominator, the additive operations simply get restricted to the operations in the numerator, under the umbrella of the common denominator.

When to find the least common denominator

Like we mentioned before, you will be interested in finding the least common denominator whenever you are computing fraction operations, especially additive operations of fractions that come with different denominators.

Having common denominators is a way of setting all the fractions in a common ground.

Example: Finding Common Denominators

Find the LCD of the numbers: 4,14, 16, 24

Solution:The first step required in order to compute the lowest common denominator (LCD) is to compute the prime decomposition of all the denominators provided 4, 14, 16 and 24.

\[4 = 2^2\] \[14 = 2 \cdot 7\] \[16 = 2^4\] \[24 = 2^3 \cdot 3\]

From the decompositions shown above, the simplest way to find the LCD is the following:

• First find ALL the primes that are present on at least one of the given numbers
• Then, find the maximum exponent for those primes across all numbers it belongs to the corresponding prime decomposition
• Multiply all primes found raised to the corresponding maximum exponent found for each, so to get the LCD
• Also, if there all the numbers are equal, we then will conclude that the LCD will that repeated number

The following primes are found, and they are listed with their maximum power found across all the prime decompositions:

• Prime = 2, Maximum Exponent = \(\max\{2,1,4,3\} = 4\)

• Prime = 7, Maximum Exponent = 1

• Prime = 3, Maximum Exponent = 1

Calculation of the Least Common Denominator (LCD)

Multiplying the all primes and their found maximum exponents, we compute the LCD as follows:

\[ LCD = \displaystyle 2^4 \cdot 7^1 \cdot 3^1 = 336 \]

This completes the calculation, and we conclude that the Least Common Denominator of the given denominators is \(LCD(4,14,16,24) = 336 \).

which concludes the calculation.

Other useful fraction calculators

This least common denominator calculator is really useful for different types of fraction calculations, though in most cases for simple numbers you can do the calculations mentally.

Using a common denominator calculator is strongly rooting in the ability of finding a prime decomposition, which it is an easy but potentially laborious process.