We all know multiplication tables as we always use them to solve math problems. And, when we use these tables, we are also using **multiples**. You see, when we multiply two numbers, the answer is their **multiple.**

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## What are Multiples?

Multiplying a number by counting numbers gives us its multiples. In math, the meaning of a multiple is the product or result of one number multiplied by another number.

Let us try and understand this concept using a few examples. It is quite simple. We can get the multiples of 6 and 7 by multiplying them by numbers 1, 2, 3, …, and so on.

**The multiples of 6 are:** 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

For instance, let us take 5 ×* *6 = 30. Here 30 is a multiple of both 5 and 6.

Let us try another example.

**The multiples of 7 are:** 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …

So, what numbers would you multiply to get 14? You will use 2 and 7.

For 2 × 7, the answer is the number 14. So, 14 here is a multiple of 2 and 7.

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## Properties of Multiples

- The number of multiples of any given number is infinite.
- Every number is a
- The multiple of a number is greater than or equal to the number itself (except for 0).

You have a lot in common with your friends, don’t you? Now, numbers, too, sometimes have things in common. One of these things is multiples. When that happens, we say that the numbers have **common multiples**. And that is the topic we are going to focus on. So, let us dive in.

## What are Common Multiples?

A common multiple is defined as a whole number, a shared multiple of each set of numbers. The multiples common to two or more numbers are called the common multiples of those numbers.

Let us mark the multiples of 6 and 7 on a hundred grid. We will mark the multiples of 6 by a circle and multiples of 7 by a cross.

The numbers that are circled as well as crossed are the common multiples of 6 and 7.

So, the common multiples of 6 and 7 are 42 and 84.

We can find the common multiples of two or more numbers by listing the multiples of each number.

## What is a Least Common Multiple?

The **smallest common multiple** of two or more numbers is called the **Least** **Common Multiple** **(LCM)**.

For example, to find the common multiples of 3 and 4, we list their multiples and then find their common multiples.

**The multiples of 3 are**: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …

**The multiples of 4 are:** 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

The common multiples of 3 and 4 are: 12, 24, 36, ….

The smallest of these is 12. So the **LCM** of 3 and 4 is 12.

## Fun** Fact:**

- A number can have an infinite (unlimited) number of multiples. Therefore, any two numbers or set of numbers can have an infinite number of common multiples.

## Solved Examples

**Q1. What are the multiples of the number 9?**

**Solution:**

We know that we can get multiples of a number by multiplying it by 1, 2, 3, …, and so on. So, the multiples of 9 are : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …

**Q2. Find two common multiples of the numbers 2 and 10.**

**Solution:**

We know that the multiples common to two or more numbers are called the common multiples of those numbers.

Now, the multiples of 2 are: 2, 4, 6, 8,** 10,** 12, 14, 16, 18,** 20, **… and the multiples of 10 are: **10**, **20**, 30, 40, 50, 60, 70, 80, 90, 100, …

So, the two common multiples of 2 and 10 are 10 and 20.

**Q3. Find the LCM of 3 and 5.**

**Solution:**

The smallest common multiple of two or more numbers is called the Least Common Multiple (LCM). The multiples of 3 are: 3, 6, 9, 12, **15**, 18, 21, 24, 27, **30,** …

The multiples of 5 are: 5, 10, **15,** 20, 25, **30**, 35, 40, 45, 50, … The common multiples of 3 and 5 are: 15, 30, …

The smallest of these is 15. So the LCM of 3 and 5 is 15.

## Practice Problems

## Common Multiples - Definition with Examples

### Which of the following is not a multiple of 8?

The multiples of 8 are: 8, 16, 24, 32, 40, …

So, 28 is not a multiple of 8.

### Which of the following is a common multiple of 16 and 20?

Now,the multiples of 16 are$\colon$16,32,48,64,$\underline{80}$,96,112,128,144,$\underline{160}\dots$

and the multiples of 20 are$\colon$20,40,$\underline{60}$,80,100,120,140,$\underline{160}$,180,200$\dots$

So,the common multiples of 16 and 20 are$\colon$80 and 160,of which 80 is option d.

### Which of the following is not a common multiple of 12 and 15?

30 is a multiple of 15, but not of 12. So, 30 is not a common multiple of 12 and 15.

### Which of the following is the LCM of 9 and 18?

18 is the smallest common multiple of 9 and 18. So, 18 is the LCM of 9 and 18.

## Frequently Asked Questions

**What are the multiples of the number zero?**

The multiples of zero is zero. Every other whole number has infinitely many multiples.

For example: 25 × 0 = 0 ; 1.0836 × 0 = 0 ; -9/87 × 0 = 0.

**What are the multiples of the number one?**

All the natural numbers are multiples of 1. There is no end to multiples of any number. The first ten multiples of the number 1 starting from 1 are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

**What is the LCM of prime numbers?**

The short form LCM stands for Least Common Multiple. The smallest common multiple of two or more prime numbers is their product.

For example: The LCM of 3 and 7 is 21.