# How to Find the Least Common Denominator

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## Introduction: How to Find the Least Common Denominator

Follow these instructions to find the Least Common Denominator for any set of fractions.

These instructions assume the reader is familiar with the basic concepts of addition, subtraction, multiplication, division, and the proper order of mathematical operations (i.e. Paretheses first, then exponents, then multiplication or division, then addition or subtraction)

## Step 1: Definitions

First we must define a few terms.

A FRACTION is a ratio of two numbers. For example 1/2, or x/(y^2)

The NUMERATOR is the number that is getting divided, it is the number on the top of the Fraction.

The DENOMINATOR is the number that is dividing the numberator, it is the number on the bottom of the Fraction.

A PRODUCT is the result of multiplying a set of numbers together. For example in the equation: 7*3*5 = 105, 105 is the Product of 7, 3 and 5.

A FACTOR is part of a Product of numbers. For example in the equation: 7*3*5 = 105, 7 is a Factor of the Product 105, 3 is a Factor of the Product 105, and 5 is a Factor of the Product 105.

A UNIQUE FACTOR is a Factor that only occurs once in a Product and a NON-UNIQUE Factor is a Factor that occurs more than once in a Product. For example in the equation 2*2*2*3*7 = 168, 3 and seven are Unique Factors of the Product 168, while 2 is a Non-Unique factor because it occurs more than once in the Product.

An EXPONENT represents the number of times a Factor is multiplied by itself. For example in the equation 2^3 = 8, 3 is the Exponent of 2. An Exponent of 3 means 2 is multiplied by itself 3 times to equal 8, i.e. 2*2*2 = 8.

The COMMON DENOMINATOR of a set of fractions is simply the Product of all the denominators. For Example, the Common Denominator of (1/4) and (1/8) is 32, which is the Product of 8 and 4.

The smallest Denominator that each fraction can have in common is the LEAST COMMON DENOMINATOR.

## Step 2: A Set of Fractions

TO FIND THE LEAST COMMON DENOMINATOR FOLLOW THE STEPS BELOW

Suppose we have a set of Fractions. For example say we have the Fractions:

1/2, 1/(7), 1/(7^3), 1/(5^3), 1/(5^4), and 1/3.

Each Denominator can be considered a Factor of the Common Denominator, which is the Product of all the Denominators. For example, for this set of Fractions the Common Denominator will be given by the following equation:

Common Denominator = 2*7*(7^3)*(5^3)*(5^4)*3

## Step 3: Identify the UNIQUE FACTORS of the Common Denominator.

In this case, the Unique Factors are 2 and 3 because they only occur once in the Product.

## Step 4: Identify NON-UNIQUE Factors in the Common Denominator

In this case 5 and 7 are Non-Unique Factors because they occur more than once in the Product. For example, 7 occurs four times in the product, once as 7, and three more times as (7^3). (Recall (7^3) = 7*7*7)

## Step 5: Find the LEAST COMMON DENOMINATOR

The LEAST COMMON DENOMINATOR is defined as the Product of (1) the Unique Factors and (2) the Non-Unique Factors which are raised to the highest Exponent.

For example, the Least Common Denominator of the set of Fractions we started with will be as follows:

LCD= (product of unique factors)*(product of non-unique factors which are raised to the highest exponent)

LCD = (2*3)*((7^3)*(5^4))