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

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

Question:

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(26 and 91)

Given:

Integers: 26 and 91

To Find:

  • LCM of 26 and 91
  • HCF of 26 and 91
  • Verify that LCM × HCF = 26 × 91

Formula:

For any two positive integers \( a \) and \( b \):

$$ \mathrm{LCM}(a, b) \times \mathrm{HCF}(a, b) = a \times b $$

Solution:

Step 1: Write the prime factorization of each number.

$$ 26 = 2 \times 13 \\ 91 = 7 \times 13 $$

Step 2: Express the prime factorization using exponents.

$$ 26 = 2^1 \times 13^1 $$ $$ 91 = 7^1 \times 13^1 $$

Step 3: Find the HCF by taking the common prime factors with the lowest exponent.

The common prime factor is 13.

$$ \mathrm{HCF}(26, 91) = 13^1 = 13 $$

Step 4: Find the LCM by taking all prime factors with the highest exponent.

$$ \mathrm{LCM}(26, 91) = 2^1 \times 7^1 \times 13^1 $$

Step 5: Calculate the value of the LCM.

$$ 2 \times 7 = 14 $$ $$ 14 \times 13 = 182 $$ $$ \mathrm{LCM}(26, 91) = 182 $$

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

$$ \mathrm{LCM}(26, 91) \times \mathrm{HCF}(26, 91) = 182 \times 13 = 2366 \\ 26 \times 91 = 2366 $$

Since both products are equal, the relation is verified.

Result:

The LCM of 26 and 91 is 182.
The HCF of 26 and 91 is 13.
$$ \mathrm{LCM}(26, 91) \times \mathrm{HCF}(26, 91) = 2366 = 26 \times 91 $$
The relation is verified.

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