Question

There are two numbers which are greater than 21 and their LCM and HCF are 3003 and 21 respectively. What is the sum of these numbers?

• A. 504
• B. 508
• C. 514
• D. 528

Answer 1

The Correct Option is A

Let the two numbers be \( x \) and \( y \). From the property of LCM and HCF, we know:

\[ \text{LCM}(x, y) \times \text{HCF}(x, y) = x \times y \]

Substituting the given values:

\[ 3003 \times 21 = x \times y \]

\[ x \times y = 63063 \]

Let \( x = 21a \) and \( y = 21b \), where \( a \) and \( b \) are coprime. Hence:

\[ \text{LCM}(x, y) = 21 \times \text{LCM}(a, b) = 3003 \]

\[ \text{LCM}(a, b) = \frac{3003}{21} = 143 \]

Now, \( a \times b = \frac{x \times y}{21^2} = \frac{63063}{441} = 143 \).

Thus, the numbers \( a \) and \( b \) are the factors of 143, which are 11 and 13.

Therefore, \( x = 21 \times 11 = 231 \) and \( y = 21 \times 13 = 273 \). The sum of the numbers is:

\[ 231 + 273 = 504 \]