Find the least perfect square that is divisible by 21, 36 and 66.

Correct option is A

Given:

We are tasked with finding the least perfect square that is divisible by the numbers 21, 36, and 66.

Concept Used:

To find the least perfect square divisible by multiple numbers, we first calculate the LCM (Least Common Multiple) of the given numbers. After that, we adjust the LCM by adding any necessary factors to make it a perfect square.

Solution:

Prime factorization of the given numbers.

​$$21 = 3 \times 7 \\36 = 2^2 \times 3^2 \\66 = 2 \times 3 \times 11$$

the LCM of 21, 36, and 66 by taking the highest powers of all prime factors involved.

​$$LCM = 2^2 \times 3^2 \times 7 \times 11 = 2772$$​​

​The LCM to make it a perfect square

The prime factorization of 2772

For a number to be a perfect square, all the powers in its prime factorization must be even.
To make the powers of 7 and 11 even, we multiply by 7 and 11.

Thus, the least perfect square = 2772 × 7 × 11 = 213444

The least perfect square divisible by 21, 36, and 66 is 213444.