HCF (Highest Common Factor) and LCM (Least Common Multiple)

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1. Highest Common Factor (HCF)

Definition

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of the given numbers exactly. It is also known as the Greatest Common Divisor (GCD).

Methods to Find HCF

  1. Prime Factorization Method
  2. Division Method
  3. Listing Factors Method

Example 1: Find HCF of 24 and 36 using Prime Factorization

  1. Prime factorization of 24 = 2 × 2 × 2 × 3
  2. Prime factorization of 36 = 2 × 2 × 3 × 3
  3. Common factors = 2 × 2 × 3 = 12
    ✅ HCF(24, 36) = 12

Example 2: Find HCF of 72 and 120 using Division Method

  1. Divide the larger number by the smaller number:
    • 120 ÷ 72 = 1 remainder 48
  2. Now divide 72 by 48:
    • 72 ÷ 48 = 1 remainder 24
  3. Now divide 48 by 24:
    • 48 ÷ 24 = 2 remainder 0
  4. ✅ HCF(72, 120) = 24

2. Least Common Multiple (LCM)

Definition

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the given numbers.

Methods to Find LCM

  1. Prime Factorization Method
  2. Listing Multiples Method
  3. Division Method

Example 1: Find LCM of 12 and 18 using Prime Factorization

  1. Prime factorization of 12 = 2 × 2 × 3
  2. Prime factorization of 18 = 2 × 3 × 3
  3. LCM is found by taking the highest powers of all factors:
    • LCM = 2 × 2 × 3 × 3 = 36 ✅ LCM(12, 18) = 36

Example 2: Find LCM of 6, 8, and 12 using Listing Multiples

  1. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
  2. Multiples of 8: 8, 16, 24, 32, 40, 48, …
  3. Multiples of 12: 12, 24, 36, 48, …
  4. ✅ LCM(6, 8, 12) = 24 (First common multiple)

3. Relationship Between HCF and LCM

For any two numbers, the product of their HCF and LCM is equal to the product of the two numbers.

HCF(a,b)×LCM(a,b)=a×bHCF(a,b)×LCM(a,b)=a×b

Example:

For 8 and 12:

  • HCF(8, 12) = 4
  • LCM(8, 12) = 24
  • HCF × LCM = 4 × 24 = 8 × 12 = 96 ✅

4. Practice MCQs on HCF and LCM

1. What is the HCF of 16 and 24?

a) 4

b) 6

c) 8

d) 12

✅ Answer: c) 8

2. What is the LCM of 5 and 7?

a) 12

b) 35

c) 70

d) 25

✅ Answer: b) 35

3. What is the HCF of 9 and 27?

a) 3

b) 9

c) 27

d) 18

✅ Answer: b) 9

4. The LCM of 10 and 15 is:

a) 20

b) 30

c) 40

d) 50

✅ Answer: b) 30

5. If HCF of two numbers is 5 and LCM is 60, and one number is 15, what is the other number?

a) 20

b) 25

c) 30

d) 45

✅ Answer: c) 20 (Since HCF×LCM=15×XHCF×LCM=15×X)

6. Which of the following numbers has the highest LCM with 4?

a) 8

b) 6

c) 12

d) 16

✅ Answer: c) 12

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