What is the Least Common Multiple? Definition, Examples

What is LCM

The Least Common Multiple (LCM) is a cornerstone in number theory, representing the smallest positive integer that is a multiple of two or more numbers. Denoted as \(\text{LCM}(a, b)\) for two integers a and b, or \(\text{LCM}(a, b, c, \ldots)\) for more than two, it encapsulates the essence of shared multiples.

Example of LCM Calculation:
Consider finding the LCM of 15 and 20.

  1. Listing Multiples:
    • List the multiples of each number: Multiples of 15 (15, 30, 45, …) and multiples of 20 (20, 40, 60, …).
  2. Identifying Common Multiples:
    • Identify the common multiples: 60 is the smallest common multiple.
  3. LCM Calculation:
    • Hence, LCM(15,20)=60.

Properties of LCM:

  1. Relation to HCF: Bézout’s Identity harmonizes LCM and HCF through \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
  2. Multiples and Factors: The LCM is the smallest multiple of each given number and is the product of the numbers divided by their HCF.
  3. Prime Factorization: Ascertain the LCM by considering the highest powers of prime factors within the given numbers.

How to Find LCM?

Method 1: Listing Multiples

Example: 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 \(24\).

Method 2: Prime Factorization

Example: Finding LCM of 9 and 15

  1. Prime Factorize Each Number:
    • Prime factorization of 9: \(3^2\).
      Prime factorization of 15: \(3 \times 5\).
  2. Identify the Factors:
    • Include unique prime factors with their highest powers:\( 3^2\) and \( 5\).
  3. Multiply the Factors:
    • Multiply the selected factors: \(3^2 \times 5\) = \(45\).
  4. Result:
    • The LCM of \(9\) and \(15\) is \(45\).

Difference Between LCM and HCF

Aspect LCM (Least Common Multiple) HCF (Highest Common Factor)
Definition The smallest positive multiple common to two or more numbers. The largest positive factor common to two or more numbers.
Representation LCM(a, b) or LCM(a, b, c, …) HCF(a, b) or HCF(a, b, c, …)
Calculation Methods
  • Listing Multiples
  • Prime Factorization
  • Bézout’s Identity
  • Listing Factors
  • Prime Factorization
  • Bézout’s Identity
Result A common multiple of the given numbers. A common factor of the given numbers.
Relationship Related to finding a common multiple for diverse applications. Related to finding a common factor, often for simplification or divisibility tests.
Notation Example LCM(4, 6) = 12 HCF(4, 6) = 2

LCM Examples

Example 1: Finding LCM of 7 and 11
  1. List Multiples:
    • Multiples of 7: 7, 14, 21, 28, 35, …
    • Multiples of 11: 11, 22, 33, 44, 55, …
  2. Identify the Common Multiple:
    • The common multiple is 77.
  3. Result:
    • The LCM of 7 and 11 is 77.
Example 2: Finding LCM of 14 and 21
  1. List Multiples:
    • Multiples of 14: 14, 28, 42, 56, 70, …
    • Multiples of 21: 21, 42, 63, 84, 105, …
  2. Identify the Common Multiple:
    • The common multiple is 42.
  3. Result:
    • The LCM of 14 and 21 is 42.
Example 3: Finding LCM of 6 and 9
  1. List Multiples:
    • Multiples of 6: 6, 12, 18, 24, 30, …
    • Multiples of 9: 9, 18, 27, 36, 45, …
  2. Identify the Common Multiple:
    • The common multiple is 18.
  3. Result:
    • The LCM of 6 and 9 is 18.
Example 4: Finding LCM of 5 and 25
  1. List Multiples:
    • Multiples of 5: 5, 10, 15, 20, 25, …
    • Multiples of 25: 25, 50, 75, 100, 125, …
  2. Identify the Common Multiple:
    • The common multiple is 25.
  3. Result:
    • The LCM of 5 and 25 is 25.