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.
 Listing Multiples:
 List the multiples of each number: Multiples of 15 (15, 30, 45, …) and multiples of 20 (20, 40, 60, …).
 Identifying Common Multiples:
 Identify the common multiples: 60 is the smallest common multiple.
 LCM Calculation:
 Hence, LCM(15,20)=60.
Properties of LCM:
 Relation to HCF: Bézout’s Identity harmonizes LCM and HCF through \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
 Multiples and Factors: The LCM is the smallest multiple of each given number and is the product of the numbers divided by their HCF.
 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
 List Multiples:
 Multiples of \(8: 8, 16, 24, 32, 40,\) …
 Multiples of \(12: 12, 24, 36, 48, 60\), …
 Identify the Common Multiple:
 The common multiple is \(24\).
 Result:
 The LCM of \(8\) and \(12\) is \(24\).
Method 2: Prime Factorization
Example: Finding LCM of 9 and 15
 Prime Factorize Each Number:
 Prime factorization of 9: \(3^2\).
Prime factorization of 15: \(3 \times 5\).
 Prime factorization of 9: \(3^2\).
 Identify the Factors:
 Include unique prime factors with their highest powers:\( 3^2\) and \( 5\).
 Multiply the Factors:
 Multiply the selected factors: \(3^2 \times 5\) = \(45\).
 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 


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