NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 4

NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 4

Question:

Given that HCF (306, 657) = 9, find LCM (306, 657).

Given:

HCF (306, 657) = 9

To Find:

LCM (306, 657)

Formula:

For any two positive integers \(a\) and \(b\):
$$ \gcd(a, b) \times \mathrm{LCM}(a, b) = a \times b $$

Solution:

Step 1: Let \(a = 306\) and \(b = 657\). The HCF is given as 9.

$$ \gcd(306, 657) = 9 $$

Step 2: Write the HCF-LCM relationship for two numbers.

$$ \gcd(306, 657) \times \mathrm{LCM}(306, 657) = 306 \times 657 $$

Step 3: Substitute the known value of HCF into the formula.

$$ 9 \times \mathrm{LCM}(306, 657) = 306 \times 657 $$

Step 4: Rearrange to solve for the LCM.

$$ \mathrm{LCM}(306, 657) = \frac{306 \times 657}{9} $$

Step 5: Calculate the product \(306 \times 657\).

$$ 306 \times 657 = 201{,}042 $$

Step 6: Substitute the product into the LCM formula.

$$ \mathrm{LCM}(306, 657) = \frac{201{,}042}{9} $$

Step 7: Divide 201,042 by 9 to get the LCM.

$$ \mathrm{LCM}(306, 657) = 22{,}338 $$

Result:

The LCM of 306 and 657 is:
$$ \mathrm{LCM}(306, 657) = 22{,}338 $$

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