NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 2(iii)

NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 2(iii)

Question:

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF equals the product of the two numbers.
336 and 54

Given:

Two integers: 336 and 54

To Find:

  • The LCM and HCF of 336 and 54.
  • Verify that LCM × HCF = product of the two numbers.

Formula:

  • Prime factorization is used to find LCM and HCF.
  • $$ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} $$

Solution:

Step 1: Find the prime factorization of 336.

$$ 336 \div 2 = 168 $$ $$ 168 \div 2 = 84 $$ $$ 84 \div 2 = 42 $$ $$ 42 \div 2 = 21 $$ $$ 21 \div 3 = 7 $$
Thus, $$ 336 = 2^4 \times 3^1 \times 7^1 $$

Step 2: Find the prime factorization of 54.

$$ 54 \div 2 = 27 $$ $$ 27 \div 3 = 9 $$ $$ 9 \div 3 = 3 $$ $$ 3 \div 3 = 1 $$
Thus, $$ 54 = 2^1 \times 3^3 $$

Step 3: Find the HCF (Highest Common Factor) of 336 and 54.

Take the lowest power of each common prime factor:

  • For 2: minimum exponent is 1
  • For 3: minimum exponent is 1
$$ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 $$

Step 4: Find the LCM (Lowest Common Multiple) of 336 and 54.

Take the highest power of each prime factor present:

  • 2: highest exponent is 4
  • 3: highest exponent is 3
  • 7: highest exponent is 1
$$ \text{LCM} = 2^4 \times 3^3 \times 7^1 $$

Step 5: Calculate the value of the LCM.

$$ 2^4 = 16 $$ $$ 3^3 = 27 $$ $$ 16 \times 27 = 432 $$ $$ 432 \times 7 = 3024 $$ $$ \text{LCM} = 3024 $$

Step 6: Verify that LCM × HCF equals the product of the two numbers.

$$ \text{LCM} \times \text{HCF} = 3024 \times 6 = 18144 $$ $$ 336 \times 54 = 18144 $$

Since both are equal, the property is verified.

Result:

The LCM of 336 and 54 is 3024, and the HCF is 6.

Verification:
$$ \text{LCM} \times \text{HCF} = 3024 \times 6 = 18144 $$ $$ 336 \times 54 = 18144 $$
Therefore, LCM × HCF equals the product of the two numbers.

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