What is the Least Common Multiple? Definition, Examples

What is LCM

The Least Common Multiple (LCM) is a fundamental concept in number theory, representing the smallest positive integer that is a multiple of two or more numbers. For two integers aa and bb, the LCM is denoted as \(\text{LCM}(a, b)\). In the case of more than two numbers, it is extended to \(\text{LCM}(a, b, c, \ldots)\).

Example of LCM Calculation: Let’s find the LCM of 12 and 18.

1. Listing Multiples:
   • List the multiples of each number: Multiples of 12 (12, 24, 36, 48, …) and multiples of 18 (18, 36, 54, …).
2. Identifying Common Multiples:
   • Identify the common multiples: 36 is the smallest common multiple.
3. LCM Calculation:
   • Thus,  \(\text{LCM}(12, 18) = 36\).

Properties of LCM:

• Relation to HCF: Bézout’s Identity connects the LCM and HCF through the equation \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\).
• Multiples and Factors: The LCM is the smallest number that is a multiple of each given number, and it is also the product of the numbers divided by their HCF.
• Prime Factorization: The LCM can be found by considering the highest powers of prime factors present in the given numbers.

How to Find LCM?

Finding the Least Common Multiple (LCM) involves various methods, and here are a few common approaches:

Method 1: Listing Multiples

1. Identify the Numbers:
   • Consider the numbers for which you want to find the LCM.
2. List Multiples:
   • Write down the multiples of each number until you find a common multiple.
     • For example, if you want to find the LCM of 4 and 6:
       • Multiples of 4: 4, 8, 12, 16, 20, …
       • Multiples of 6: 6, 12, 18, 24, …
3. Identify the Common Multiple:
   • Find the smallest common multiple. In this case, it’s 12.
4. Result:
   • The LCM of 4 and 6 is 12.

Method 2: Prime Factorization

1.Prime Factorize Each Number:

Decompose each number into its prime factors.
Example: Prime factorization of 12 = \(2^2 \times 3\), and

prime factorization of 18 = \(2 \times 3^2\).

2. Identify the Factors:

Include all the unique prime factors with their highest powers.
For LCM(12, 18), include \(2^2\) and \(3^2\).

3. Multiply the Factors:

Multiply the selected factors.

\(2^2 \times 3^2\) = \(4 \times 9 \)= 36

4. Result:

The LCM of 12 and 18 is 36.

Difference Between LCM and HCF

Let’s present the differences between the Least Common Multiple (LCM) and Highest Common Factor (HCF)..

LCM Examples

Example 1: Finding LCM of 8 and 12

1. List Multiples:
   • Multiples of 8: 8, 16, 24, 32, 40, …
   • Multiples of 12: 12, 24, 36, 48, 60, …
2. Identify the Common Multiple:
   • The common multiple is 24.
3. Result:
   • The LCM of 8 and 12 is 24.

Example 2: Finding LCM of 10 and 15

1. List Multiples:
   • Multiples of 10: 10, 20, 30, 40, 50, …
   • Multiples of 15: 15, 30, 45, 60, 75, …
2. Identify the Common Multiple:
   • The common multiple is 30.
3. Result:
   • The LCM of 10 and 15 is 30.

Example 3: Finding LCM of 18 and 24

1. List Multiples:
   • Multiples of 18: 18, 36, 54, 72, 90, …
   • Multiples of 24: 24, 48, 72, 96, 120, …
2. Identify the Common Multiple:
   • The common multiple is 72.
3. Result:
   • The LCM of 18 and 24 is 72.

Example 4: Finding LCM of 16 and 20

1. List Multiples:
   • Multiples of 16: 16, 32, 48, 64, 80, …
   • Multiples of 20: 20, 40, 60, 80, 100, …
2. Identify the Common Multiple:
   • The common multiple is 80.
3. Result:
   • The LCM of 16 and 20 is 80.

Example 5: Finding LCM of 9 and 14

1. List Multiples:
   • Multiples of 9: 9, 18, 27, 36, 45, …
   • Multiples of 14: 14, 28, 42, 56, 70, …
2. Identify the Common Multiple:
   • The common multiple is 126.
3. Result:
   • The LCM of 9 and 14 is 126.