Question

The HCF and the LCM of two numbers is 21 and 4641, respectively. If one of the numbers, lies between 200 and 300, then the two numbers are:
A. 273,363
B. 273,359
C. 273,361
D. 273,357

Answer 1

Hint: It is known that the product of the HCF of the numbers and the LCM of the numbers is equal to the product of those numbers. So, use this formula to find the product in such a way that one of them satisfied the given condition of lying in between 200 and 300.

Complete step-by-step solution
Let us take the two numbers to be x and y.
Consider the HCF and LCM of the two numbers, which are 21 and 4641. We use the following formula to find the product of the two numbers.
$
  {\text{HCF}} \times {\text{LCM}} = x \times y \\
   \Rightarrow 21 \times 4641 = x \times y \\
$

Now, we look at the given options, in each of the given options one number is 273, we make a manipulation in such a way that we get a product of 273 and the second number. Let us factor both 21 and 4641.
$ \Rightarrow \left( {3 \times 7} \right) \times \left( {3 \times 7 \times 13 \times 17} \right) = x \times y$
Now we combine and multiply the factor that gives the product as 273 and combine and multiply the remaining to get the other number.
$
   \Rightarrow \left( {21 \times 13} \right) \times \left( {21 \times 17} \right) = x \times y \\
   \Rightarrow 273 \times 357 = x \times y \\
$

Thus, we may conclude that one number is 273 and the other is 357.

Hence, option (D) is the correct option.

Note: In the question having the options and a condition where many answers are possible, we begin by looking at the options first and analysing if they have any value in common. If yes, we try to fix that value and solve the question with respect to that value, as in this case it was 273. Even more than one answer would be valid for this question, but in order to match the options, we fixed ourselves to the number 273.