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# Least Common Denominator|Definition & Meaning

## Definition

The **least common denominator **is the smallest common **multiple **of the denominators of two or more fractions. For example, for **two** fractions, 1/3 and 3/2, the **smallest **common multiple of the **denominators**, 3 and 2, is 6, and hence it is the **least common denominator**. The least common denominator is used during the addition and subtraction of **fractions**.

A **value **used to determine the **smallest **number that two or more **denominators**—the bottom value in a **fraction**—have in **common **is the least common denominator, often referred to as the least common multiple. In **mathematics**, especially when dealing with **fractions**, this concept is fundamental.

## Use of LCD

Various **fractions **with different **denominators **are summed or subtracted using the least common denominator (LCD). To **subtract** or **add** two or more fractions, a **common denominator** is required. This is so that the **fractions **can never be added together since the **denominators **represent the **units **in which the **fraction **is measured.

To make all the **denominators **equal and enable the **addition **or **subtraction **of the fractions, there must be a common denominator for two or more fractions.

For instance, we must first find out the **least common denominator **before adding the fractions **1/2** and **1/3**. The LCD is **6** because it is the least common multiple of **2** and **3**.

Then, we can rewrite both **fractions **using a **6** as the **denominator**: **1/3** turns into **2/6** and **1/2** into **3/6**. Now that the **denominators **of the two **fractions **are the same, we can combine them:

3/6 + 2/6 = 5/6

## Methods of Finding the Least Common Denominator

There are commonly **two **methods to find the **least common denominator**. They are discussed as follows.

### The Common Method

Usually, to determine the **least common denominator**, one must find the **least common multiple** of the denominators. We can list the **multiples **of each number and identify the one that the two **numbers **have in **common **the least.

If there are more than **two** numbers, we can calculate the **LCM** of the first two before calculating the **LCM** of the outcome and the following number, and so on, until we have calculated the **LCM** of every number.

For example, to find the **LCM** of **4** and **6**, we list out the **multiples **of each:

4, 8, 12, 16, 20, 24, 28, 32, 36, and so on are **multiples** of **four**.

6, 12, 18, 24, 30, 36, 42, 48, 54, and so on are **multiples** of **six**.

The smallest **multiple **that both **4** and **6** have in **common **is **12**, so the **LCM** of **4** and **6** is **12**.

Using the same procedure, we can determine the **LCM **of three or more numbers by finding the **LCM** of the result and the subsequent **number **until we have determined the **LCM **of all the numbers.

For instance, we can first determine the **LCM **of **4** and **6**, which is **12**, before determining the **LCM **of **4**, **6**, and **8**. The **LCM** of **12** and **8 **is then discovered to be **24**. Thus, 24 is the **LCM** of 4, 6, and 8.

### Prime Factorization Method

**Prime factorization **can also be used to get the **LCM**. Finding the prime factorization of each **integer** allows us to determine the **LCM **of two numbers by **multiplying **the greatest **power **of each **prime factor**. For instance, we can first determine the prime factorization of each to determine the LCM of 15 and 20:

15 = 3 * 5

20 = 2 * 2 * 5

The **LCM** is produced by **multiplying **each prime **factor **by its **maximum **power:

LCM of 15 and 20 is equal to 2 * 2 * 3 * 5 = 60.

By following the same procedure, finding the **LCM **of the outcome and the subsequent number until we have discovered the **LCM **of all the numbers, we can determine the LCM of more than **two** numbers.

## Properties of Least Common Denominator

In the field of **mathematics, **apply the idea of the **least common denominator** (LCD) to add or subtract **fractions **with various **denominators**. It is the denominator of two or more fractions with the smallest common **multiple**. The least common denominator has the following characteristics:

- Uniqueness: The
**LCD**of a collection of**fractions**is**unique**, which means that for a given set of**denominators**, there is only one**smallest**common multiple. - Multiplication: By
**multiplying**the**fractional**denominators, the**LCD**can be calculated. - Divisibility: Each of the fractions’
**denominators**must be able to divide the**LCD**. - Positive Value: The
**LCD**must always have a**positive**number. - Monotonicity: As the
**number**of**fractions**in a**set**of fractions increases, so does the**LCD**. - Commutativity: It makes no difference what
**order**the**denominators**are**multiplied**to produce the**LCD**. - Associativity: The
**LCD**of a**set**of**fractions**is**associative**, which means that the result is unaffected by the arrangement of the**denominators**. - Transitivity: The
**LCD**of all three**denominators**is the case where a**common**multiple of two**denominators**is also a**multiple**of a**third**denominator.

The **LCD **is not necessarily the **result **of the **denominators;** it is vital to remember this. Given the requirement that the **LCD **be the **smallest common multiple**, it can be a **multiple** of numbers apart from the **denominators** along with the **LCD**.

It is necessary to convert each **fraction **to an equal fraction with the same **denominator**, which is the **LCD**, in order to **add **or **subtract **fractions. As a result, we can add and subtract the **fractions **as if they had the same **denominator**. When it comes to adding or subtracting fractions with various **denominators**, the **LCD **is a useful technique.

The **least common denominator **is a useful **mathematical **concept that enables us to add and subtract fractions with various denominators. Its properties, such as uniqueness, divisibility, and positivity, make it a valuable resource for tackling **fraction-related **mathematical problems.

## Solved Examples Involving the Least Common Denominator

### Example 1

Calculate the least common denominator of 1/4 and 1/6.

### Solution

These two **fractions** have the **denominators **4 and 6, respectively. **4 **and **6** have the following **multiples**:

**Multiples **of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, and so on.

**Multiples **of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, and so on.

We can observe from this list of **multiples **that **12 **is the **least common multiple**. Therefore, these two fractions’ least common denominator is twelve.

### Example 2

Find the **least common denominator **of **1/8 **and **1/12** by using the method of **prime factorization**.

### Solution

These **fractions **have **denominators **of **8 **and **12**.

The **prime factorizations **of these two **denominators **must now be determined:

8 has the following prime factors: 2 x 2 x 2

12 has the following prime factors: 2 x 2 x 3

The **prime numbers** 2 and 3 appear the most frequently: 2 x 2 x 2 in 8 and 3 in 12.

We now need to **multiply **each factor separately:

2 × 2 × 2 × 3 = 24

Thus, **24 **is the **least common denominator **of the fractions **1/8** and **1/12**.

*All images/mathematical drawings were created using GeoGebra.*