NCERT Class X Chapter 1: Real Numbers Example 2

NCERT Class X Chapter 1: Real Numbers Example 2

Question:

Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Given:

Numbers: 6 and 20

To Find:

LCM and HCF of 6 and 20

Formula:

  • Prime Factorisation Method:
    • HCF: Product of the lowest powers of all common prime factors.
    • LCM: Product of the highest powers of all prime factors present in any number.

Solution:

Step 1: Write the prime factorisation of each number.

$$ 6 = 2 \times 3 $$ $$ 20 = 2 \times 2 \times 5 = 2^2 \times 5 $$

Step 2: List all prime factors with their powers.

  • 6: \(2^1,\, 3^1\)
  • 20: \(2^2,\, 5^1\)

Step 3: Find the HCF by taking the product of the lowest powers of all common prime factors.

  • Common prime factor: 2
  • Lowest power of 2: 1
$$ \text{HCF}(6,\,20) = 2^1 = 2 $$

Step 4: Find the LCM by taking the product of the highest powers of all prime factors present.

  • 2: Highest power is 2
  • 3: Highest power is 1
  • 5: Highest power is 1
$$ \text{LCM}(6,\,20) = 2^2 \times 3^1 \times 5^1 $$

Step 5: Calculate the value of the LCM.

$$ 2^2 = 4 $$ $$ 4 \times 3 = 12 $$ $$ 12 \times 5 = 60 $$ $$ \text{LCM}(6,\,20) = 60 $$

Result:

HCF(6, 20) = 2

LCM(6, 20) = 60

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