Common Multiple

• Introduction
• What Do You Mean by Common Multiples in Mathematics?
• How To Calculate the Common Multiples
• Common Multiples Between `2` Numbers
• Common Multiples Between `3` Numbers
• Properties of Common Multiplies
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

In primary school maths, we learned about multiples. The concept of multiples is important, particularly when working with Least Common Multiples (LCM). It is also used when identifying the patterns in numbers. In this article, we will discuss Common Multiples and Least Common Multiples (LCM) in depth with appropriate examples and diagrams.

What Do You Mean by Common Multiples in Mathematics?

In mathematics, common multiples are the multiples that are shared by a particular set of numbers.

By counting every number, we can determine the multiples of a given number. A multiple in maths refers to the multiplication of two numbers. We are all familiar with multiplication tables while solving maths problems. We also find multiples while using these tables. As you can see, the result of multiplying two numbers is their multiple. We can create a list of the multiples of an integer by multiplying it by natural numbers.

Below are multiplication tables, which are representations of the first ten multiples of numbers from `1` to `10`.

For Example - In order to identify the common multiples of `8` and `9`, we can list the multiples of `8` and `9` individually and look for those that appear in both lists.

Multiples of `8` can be written as `8, 16, 24, 32, 40, 48, 56, 64, ``72`  and so on. 

Multiples of `9` can be written as `9, 18, 27, 36, 45, 54, 63, ``72``, 81` and so on.

If we look for multiples that appear in both the lists, we find `72`. This is referred to as the common multiple of `9` and `8`.

How To Calculate the Common Multiples

We can figure out the multiples of two or more integers by listing their individual multiples.

On a grid of `100`, let's indicate the multiples of `7` and `6`. The multiples of `7` are marked with a cross and the multiples of `6` are marked with a circle.

Multiples of `7: 7, 14, 21, 28, 35, ``42``, 49, 56, 63, 70, 77, ``84``, 91, 98, ....`

Multiples of `6: 6, 12, 18, 24, 30, 36, ``42``, 48, 54, 60, 66, 72, 78, ``84``, 90, 96, …..`

Common Multiples of `7` and `6:``42, 84``,....`

Common Multiples Between `2` Numbers

It is quite easy to find common multiples of two numbers. List the multiples of both numbers (starting with the first few multiples), as we mentioned in the last example. Search for the multiples that appear in both lists.

Example -  Find the common multiples of `15` and `6`.

Common multiples of `15` and `6`: `60`, …..

Common Multiples Between `3` Numbers

Common multiples of `3` numbers can be identified by using the Listing Method. Here, all we need to do is to look for multiples that are included in all three lists. In other words, we need to determine the multiples of all the `3` numbers.

Example -  Find the common multiples of `126, 14,` and `18`.

Common multiples of `126, 14` and `18`: `126`, …..

Properties of Common Multiplies

• A number has an unlimited number of multiples. As a result, any two integers or collection of numbers have an indefinite number of common multiples.
   
• The product `a\times b` of any `2` numbers, `a` and `b`, is always a common multiple of `a` and `b`.
  Example : `3\times 4` is a common multiple of `3` and `4`.
   
• When two numbers `a` and `b` are coprime, the multiples of their product are their common multiples.
  Example: `2` and `3` are coprime numbers.
  Common multiples of `2` and `3: 6, 12, 18…..`
   
• When two numbers are coprime, their product is equivalent to their LCM (Least Common Multiple).
  Example: Find the LCM of `5` & `7`?
  `5` and `7` are co-prime numbers.
  LCM`(5, 7) = 5\times 7 = 35`
   
• The LCM of `a` and `b` is `b` if and only if `b` is a multiple of `a`.
  Example: LCM`(7, 14) = 14`

Solved Examples 

Q`1`. Find the Least Common Multiple (LCM) of `6` and `9` ?

Solution:

Multiples of `6: 6, 12, ``18``, 24, 30, 36, 42, 48, 54, 60, 66, 72, …..`

Multiples of `9: 9, ``18``, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, …..`

LCM`(6, 9): ``18` 

Q`2`. Calculate the LCM of `13` & `39`.

Solution: 

The LCM of `a` and `b` is `b` if and only if `b` is a multiple of `a`.

LCM`(13, 39) = 39` because `39` is a multiple of `13`.

Q`3`. Which number is the first common multiple of `15` and `18`?

Solution: 

Multiples of `15: 15, 30, 45, 60, 75, 90, ….`

Multiples of `18 : 18, 36, 54, 72, 90,  108, …..`

First common multiple of `18` & `15: 90`

Q`4`. Which numbers are multiples of one?

Solution:

Every natural number that exists is a multiple of `1`. Any number is divisible by `1`. The first ten multiples of `1` are `1, 2, 3, 4, 5, 6, 7, 8, 9,` and `10`.

Q`5`. Calculate the LCM of `12, 18,` and `20`.

Solution: 

Multiples of `12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, ``180``….`

Multiples of `18: 18, 36, 54, 72, 90, 108, 126, 144, 162, ``180``, …..`

Multiples of `20: 20, 40, 60, 80, 100, 120, 140, 160, ``180``, 200, ....`

LCM `(12,18,20) = ``180`

Practice Problems

Q`1`. Which of the following is the LCM of `9` and `18`?

1. `18`
2. `27`
3. `36`
4. None of the above

Answer: a

Q`2`. Which number is the multiple of `15`?

1. `30`
2. `35`
3. `40`
4. None of the above

Answer: a

Q`3`. Which number is not a multiple of `12` ?

1. `36`
2. `60`
3. `72`
4. `18`

Answer: d

Q`4`. Which number is not a common multiple of `18` & `12`?

1. `36`
2. `72`
3. `114`
4. `108`

Answer: c

Q`5`. Which number is not a multiple of `25`?

1. `50`
2. `60`
3. `100`
4. `75`

Answer: b

Frequently Asked Questions

Q`1`. What is a common multiple?

Answer: A common multiple of two or more numbers is a number that is a multiple of each of the given numbers. In other words, it is a number that can be evenly divided by all the numbers in question.

Q`2`. How do you find common multiples of two numbers?

Answer: To find common multiples of two numbers, you can list the multiples of each number and identify the ones that appear in both lists. 

Q`3`. Why are common multiples important?

Answer: Common multiples are important in various mathematical concepts, such as finding equivalent fractions, adding and subtracting fractions with different denominators, and solving problems involving ratios and proportions. They are also used in real-life scenarios, such as scheduling events or coordinating tasks that occur at regular intervals.

Q`4`. What is the difference between a common multiple and a least common multiple (LCM)?

Answer: A common multiple is any multiple that is shared by two or more numbers, while the least common multiple (LCM) is the smallest common multiple of those numbers.

Q`5`. Can you have more than one common multiple for a given set of numbers?

Answer: Yes, a set of numbers can have an infinite number of common multiples. For example, the common multiples of `6` and `8` include `24, 48, 72,` and so on.