Question

The least perfect square, which is divisible by each of 21, 36 and 66 is:
$
  \left( A \right)213444 \\
  \left( B \right)214344 \\
  \left( C \right)214434 \\
  \left( D \right)231444 \\
$

Answer 1

Hint-In this question, we use the concept of least common multiple (LCM) and perfect square. Least Common Multiple (LCM) is a method to find the smallest common multiple between any two or more numbers. We have to use the definition of perfect square; it is a number that can be expressed as the product of two equal integers.

Complete step-by-step answer:
To find the least perfect square we have to find LCM of three numbers 21, 36, 66 and make LCM be a perfect square.
Now, we find LCM of 21, 36 and 66. Least Common Multiple (LCM) is a method to find the smallest common multiple between any two or more numbers.
Factor of 21 is $3 \times 7$
Factor of 36 is $2 \times 2 \times 3 \times 3$
Factor of 66 is $2 \times 3 \times 11$
Now, LCM of 21, 36 and 66 is $2 \times 2 \times 3 \times 3 \times 7 \times 11$
$ \Rightarrow {2^2} \times {3^2} \times 7 \times 11$
We can see 2 and 3 are in perfect squares but 7 and 11 are not in perfect squares so to make a perfect square we have to multiply 7 and 11 in LCM of 21, 36 and 66.
$
   \Rightarrow {2^2} \times {3^2} \times 7 \times 11 \times 7 \times 11 \\
   \Rightarrow {2^2} \times {3^2} \times {7^2} \times {11^2} \\
 $
Now all terms in perfect square so we calculate it,
$
   \Rightarrow 4 \times 9 \times 49 \times 121 \\
   \Rightarrow 36 \times 49 \times 121 \\
   \Rightarrow 213444 \\
$
So, the correct option is (A).

Note-In such types of problems we use some important points to solve questions in an easy way. First we find the LCM of three numbers given in question and then check if all numbers of LCM are in perfect square or not. If numbers of LCM are in perfect square so that is the answer of question but sometimes all numbers of LCM are not in perfect square so we multiply those numbers in LCM who are not in perfect square.