🔹 Least Common Multiple (LCM)

• Definition:
  LCM (Least Common Multiple) of two or more numbers is the smallest multiple that is common to all of them.

👉 Examples:

• LCM of 2 and 3 = 6 (smallest number divisible by both 2 and 3)

• LCM of 4, 6, 8 = 24 (smallest number divisible by all three)

🔹 Highest Common Factor (HCF)

• Definition:
  HCF (Highest Common Factor) of two or more numbers is the largest number that divides all of them exactly.

👉 Examples:

• HCF of 12 and 18 = 6

• HCF of 16 and 24 = 8

🔹 Relation Between LCM and HCF

For two numbers A and B:

LCM×HCF=A×B

✅ This relation is very useful in problem-solving.

👉 Example:
If LCM = 60 and HCF = 3, then:

Product of numbers=60×3=180 

Now, we need two numbers whose product = 180 and HCF = 3.
Numbers = 9 and 20.

📘 Examples

Example 1

Find the LCM and HCF of 12 and 20.

• Multiples of 12: 12, 24, 36, 48, 60, …

• Multiples of 20: 20, 40, 60, 80, …
  👉 LCM = 60

• Factors of 12: 1, 2, 3, 4, 6, 12

• Factors of 20: 1, 2, 4, 5, 10, 20
  👉 HCF = 4

✅ Answer: LCM = 60, HCF = 4

Example 2

Find the LCM and HCF of 15 and 25.

• Multiples of 15: 15, 30, 45, 60, 75, …

• Multiples of 25: 25, 50, 75, 100, …
  👉 LCM = 75

• Factors of 15: 1, 3, 5, 15

• Factors of 25: 1, 5, 25
  👉 HCF = 5

✅ Answer: LCM = 75, HCF = 5

Example 3

Find the LCM and HCF of 36 and 48.

• Multiples of 36: 36, 72, 108, 144, …

• Multiples of 48: 48, 96, 144, 192, …
  👉 LCM = 144

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  👉 HCF = 12

✅ Answer: LCM = 144, HCF = 12

🎯 Quick Tricks for Exams

1. Prime Factorization Method

   • LCM = Product of highest powers of all prime factors

   • HCF = Product of lowest powers of all prime factors

2. Relation Formula

   • Use LCM×HCF=Product of numbers\text{LCM} \times \text{HCF} = \text{Product of numbers} for quick calculations.

LCM and HCF for Sainik School Entrance Examination