C.M of whole numbers

JSS 1 Mathematics: Least Common Multiple (LCM) of Whole Numbers

Welcome, JSS 1 students! Today, we will learn about the Least Common Multiple, or LCM. It's a useful tool that helps us find the smallest number that is a multiple of two or more numbers. This skill is important for solving many problems, especially when working with fractions.

Core Concepts

What is a Multiple?

A multiple of a number is the result of multiplying that number by any whole number (except zero). Think of it as counting by that number. For example, multiples of 3 are 3, 6, 9, 12, 15, and so on (3x1, 3x2, 3x3, ...). Multiples of 4 are 4, 8, 12, 16, 20, ...

What are Common Multiples?

When you list the multiples of two or more numbers, any multiple that appears in all the lists is called a common multiple. Let's look at multiples of 3: {3, 6, 9, 12, 15, 18, 21, 24, ...} And multiples of 4: {4, 8, 12, 16, 20, 24, 28, ...} Here, 12 and 24 are common multiples of 3 and 4.

What is the Least Common Multiple (LCM)?

The Least Common Multiple (LCM) is the smallest of the common multiples of two or more numbers. From our example above, the common multiples of 3 and 4 are 12, 24, ... The smallest among them is 12. So, the LCM of 3 and 4 is 12.

Methods to find LCM:

1. Listing Multiples Method: This method involves listing the multiples of each number until you find the first common multiple.

   • Example: Find the LCM of 6 and 8.
     • Multiples of 6: 6, 12, 18, 24, 30, 36, ...
     • Multiples of 8: 8, 16, 24, 32, 40, ...
     • The first common multiple is 24. So, LCM (6, 8) = 24. This method is good for small numbers, but can be long for larger numbers.

2. Prime Factorization Method: This is a more efficient method, especially for larger numbers. It uses prime factors.

   • Steps:
     1. Find the prime factors of each number.
     2. Write out the prime factors using powers (e.g., 2x2x2 = 2³).
     3. For each unique prime factor, choose the one with the highest power.
     4. Multiply these highest powers together to get the LCM.
   • Example: Find the LCM of 12 and 18.
     • Step 1: Prime Factors
       • 12 = 2 x 2 x 3
       • 18 = 2 x 3 x 3
     • Step 2: Using Powers
       • 12 = 2² x 3¹
       • 18 = 2¹ x 3²
     • Step 3: Choose Highest Powers
       • For prime factor 2: The highest power is 2² (from 12).
       • For prime factor 3: The highest power is 3² (from 18).
     • Step 4: Multiply
       • LCM (12, 18) = 2² x 3² = (2 x 2) x (3 x 3) = 4 x 9 = 36.

   

   Another common way to use prime factorization is to divide the numbers by common prime factors simultaneously.

   • Example: Find LCM of 12 and 18 (using simultaneous division)
     • 2 | 12, 18
     • 3 | 6, 9
     • 2 | 2, 3
     • 3 | 1, 3
     • | 1, 1
     • LCM = 2 x 3 x 2 x 3 = 36.

   

Real-World Example

Imagine two buses. Bus A leaves the station every 15 minutes, and Bus B leaves every 20 minutes. If both buses leave at 8:00 AM, when will they next leave at the same time? We need to find the LCM of 15 and 20. (LCM of 15 and 20 is 60). So, they will both leave together again after 60 minutes, which is at 9:00 AM. This helps in scheduling! For home practice, try to find other real-life situations where you might need to find the LCM.

Practice/Reflection Question

Try to find the LCM of 9 and 15 using both the listing multiples method and the prime factorization method. Which method did you find easier, and why?