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**Common Multiple|Definition & Meaning**

**Definition**

**Common multiples** are **whole numbers** that are **shared** **among two** or **more** sets of **numbers** or a number’s common multiples correspond to the **multiples** of them that are** common** to at least two other numbers.

Several **numbers** **share** the** same multiples**. The term “common multiple” **refers** to **them**. For instance, **take 2 and 4. We** want to **compute** the **common multiple** of these two numbers. For that purpose, we will **first make** a **list** of **2 multiples**, and **secondly**, we will make a **list** of **3 multiples. The** **common multiple** will be the **number** that will **occur** in **both lists** or, in other words, is **common** in **both lists**.

**Multiples of 2** can be written as** 2, 4, 6, 8, 10, 12**, and continued. We can write **4’s multiples** as **4, 8, 12, 16, 20, 24, 28**, and so on. On figuring out the **common** values between the two lists, we got **4 and 8, and 12,** so **these** three numbers **are common multiples** of 2 as well as 4.

**Properties of Common Multiple**

- There is an
**unlimited (infinite)**common multiple of any two numbers. - The common multiple of two numbers is
**divisible**by**both numbers**. - In mathematics, a number’s common multiple is
**greater or equal to**its number. - A common multiple of two numbers will be the
**product of those two numbers**with some**other number**.

**Proof of Property 1**

There is an unlimited **(infinite) common multiple** of any two numbers.

Consider two **numbers, 2 and 4. The** **multiple of both** numbers will be **infinite** as we know that we get the **multiples of a number** by **multiplying **it with a range of numbers starting from **1 to infinity. **Therefore, **we** will **get** an **infinite list of multiples** same goes for the second number, so **we have** a **list** of **multiples of two numbers** that are **infinite,** so the **common multiple** (a common number in two lists) will** also** be **infinite**. This is shown by an illustration.

**Proof of Property 2**

The **common multiple** of two numbers is **divisible** by **both numbers**.

**Consider** two numbers, **2 and 4**. The **finite common multiple lists** of two numbers are shown. We can see **4, 8, and 12** are the **common multiple of 2 and 4** so if **we divide** them **by 2 and 4** they are **easily divisible** as shown. We can see that the **common multiple 4** **of number 2** and **number 4** is **divisible** by **both numbers.**

**Proof of Property 3**

In mathematics, a number’s **common multiple** is **greater or equal to its number**.

Consider** two numbers 2 and 4**. We can see from the **list of** their **common multiple** that** every common multiple** is **either greater** or** equal** to the number.**4, 8, and 12** are **greater** than **2 and 8, 12** is **greater** than **4** and **4 is equal to 4**. **None** of the common multiples **is less** than the **number itself.**

**Proof of Property 4**

A **common multiple** of two numbers will be the **product** of those **two numbers with** some **other number.**

Consider t**wo numbers 2 and 4.** We know that **4, 8, and 12** are the common multiple of both numbers so generally, they are the **product of the number** of **which** they are **common multiple** ( 4 and 2) **with** some **whole number.** We can clearly say that the **common multiple 4** of **number 2** and **number 4 **is** equal** to the **number itself** **multiplied** by **another** whole **number**.

**Steps to Compute Common Multiple of Two Numbers**

**Step 1: **Given 2 n**umbers, x and y,** make the list of multiple of both numbers like this.

**List1**: Multiples of **First Number** (X)

**List2**: Multiples of **Second Number** (Y)

**Step 2: ****Highlight** those multiples that are common in both list

**List1**: a, b, **c**, **d**, e, f (**Multiples of X**)

**List2**: h, i, j, k, **d**, **c **(**Multiples of Y**)

**Step 3: **Make a **new list containing** the **common numbers** in both lists.

**Common Multiple of X and Y=** c, d

**Least Common Multiple**

LCM is the **least common multiple** of the **two numbers**, **given** two numbers **x and y**, and their **multiples** as **List 1 and List 2**, **LCM** would **be** the **smallest common number** **between List 1 and List 2.** If you have more than one number, you can also calculate it. To **find** the **LCM**, there are **various** **methods** available.

For the** fast computation** of LCM of two numbers, **one approach** is to **factorize each number** and then, by **multiplying** their **highest powers**, find the **product** of the **prime factors** they have in common. The general process is detailed below:

**Step 1: ****Given** two numbers,** x and y,** **make** a l**ist of multiple** numbers like this.

**List 1**: Multiples of First Number (X)

**List 2**: Multiples of Second Number (Y)

**Step 2: ****Highlight** those **multiples** that are **common in both lists.**

**List 1**: a, **b**, c, **d**, e, f (Multiples of X)

**List 2**: h, i, **b**, k, **d**, l ** **(Multiples of Y)

**Step 3: ****Make** a **new list** containing the **common numbers** in both lists

**Common Multiple of X and Y=** b, d

**Step 4: ****Out of the list** of common multiples,** the smallest number** in that list **will be** the **LCM**.

**Properties of Least Common Multiple**

- When
**two or more prime numbers**are**multiplied**together, the**LCM****is**the**product of those numbers.** - Considering two or more numbers,
**Least common multiple**will always be**greater**than or**equal to each****number**individually.

## Solved Problems on Finding the LCM

**Example 1**

**Find 4 Common Multiples** of three numbers** 2, 4, and 8.**

**Solution**

**Step 1: **Given three numbers **2, 4, and 8.**

**List 1:** 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32

**List 2:** 4,8,12,16,20,24,28,32,26,34

**List 3:** 8,16,24,32,40,48,56,64,72,80

**Step 2: ****Highlight** those multiples that are common in both list

**List 1**: 2,4,6,**8**,10,12,14,**16**,18,20,22,**24**,26,28,30,**32**

**List 2**: 4,**8**,12,**16**,20,**24**,28,**32**,26,34

**List 3**:** 8,16,24,32**,40,48,56,64,72,80

**Step 3: ****Make** a **new list c**ontaining the common numbers in both lists

**Common Multiple of 2,4 and 8=** 8,16,24,32

**Example 2**

Find the **Least Common Multiple** of the two numbers **3** and** 6.**

**Solution**

**Step 1: **Given two numbers **3** and** 6:**

**List 1**: 3,6,9,12,15,18,21,24,27,30

**List 2**: 6,12,18,24,30,36,42,48,54,69

**Step 2: ****Highlight** those **multiples** that are **common** in both list

**List 1**: 3,**6**,9,**12**,15,**18**,21,**24**,27,**30 **

**List 2**: **6**,**12**,**18**,**24**,**30**,36,42,48,54,60

**Step 3: ****Make** a **new list** containing the **common numbers** in **both lists.**

**Common Multiple of 3 and 6**= 6,12,18,24,30

**Step 4: **The **smallest number** from the common multiple** lists** **is** 6, which is the LCM of 3 and 6.

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