UNDERSTANDING MULTIPLES OF NUMBERS - JSS2 Mathematics Lesson Note

What are Multiples?

Multiples are the result of multiplying a number by an integer. For example, the multiples of 3 are what you get when you multiply 3 by 1, 2, 3, and so on. Here's a list of the first few multiples of 3:

3×1=3

3×2=6

3×3=9

3×4=12

So, the first four multiples of 3 are 3, 6, 9, and 12. Multiples are like counting by that number.

 

Finding Multiples

To find multiples of any number, you can keep adding the number to itself. For example:

Multiples of 5: 5, 10, 15, 20, 25, ...

Multiples of 7: 7, 14, 21, 28, 35, ...

 

What is the Lowest Common Multiple (LCM)?

The Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers.

 

Why is the LCM Important?

The LCM is useful in solving problems that involve combining different cycles or schedules, like finding when two events will happen at the same time.

 

How to Find the LCM

There are a few ways to find the LCM, but let's focus on two common methods: listing multiples and prime factorization.

  Method 1: Listing Multiples

List the multiples of each number.

Find the smallest multiple that appears in both lists.

 

Example: Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Multiples of 6: 6, 12, 18, 24, 30, ...

The smallest common multiple is 12, so the LCM of 4 and 6 is 12.

  Method 2: Prime Factorization

Find the prime factors of each number.

List all prime numbers that appear in the factorizations.

For each prime number, take the highest power that appears in any of the factorizations.

Multiply these together to get the LCM.

 

Example: Find the LCM of 12 and 15.

Prime factors of 12: 12 = 2²  x  3¹

Prime factors of 15: 15 = 3¹  x  5¹

 

List all prime numbers: 2, 3, 5.

The highest power of 2 is 2².

The highest power of 3 is 3¹.

The highest power of 5 is 5¹.

 

Multiply these together to get the LCM: 2²  x  3¹  x  5¹ = 4  x  3  x  5 = 60.

So, the LCM of 12 and 15 is 60.

 

Practice Example: Find the LCM of 8 and 12

Method 1: Listing Multiples

Multiples of 8: 8, 16, 24, 32, 40, 48, ...

Multiples of 12: 12, 24, 36, 48, ...

The smallest common multiple is 24.

 

Method 2: Prime Factorization

Prime factors of 8: 8=2³

Prime factors of 12: 12=2²  x  3¹

List all prime numbers: 2, 3.

The highest power of 2 is 2³.

The highest power of 3 is 3¹.

Multiply these together to get the LCM: 2³  x  3¹ = 8  x  3 = 24.

So, the LCM of 8 and 12 is 24.