Prime factorization of\(21 = 3 \times7\)
Prime factorization of\(36 = 2 \times 2 \times 3 \times 3 \text{ or } 22 \times 32\)
Prime factorization of\(66 = 2 \times 3 \times11\)
Maximum of all the prime exponents that exist in the numbers above will be
LCM\(= 22 \times 32 \times 7 \times 11\)
→ The least number, which is divisible by 21,36, and 66 is\(22 \times 32 \times 7 \times 11\)
In a perfect square, always exponent of each prime is always even, so
In order to find the least perfect square, we will make each exponent even
\(22 \times 32 \times 72 \times 112 = 213444\)
Therefore, the least perfect square, which is divisible by 21, 36, and 66 is 213444.
The correct answer is (B): 213444
Directions: In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option from the following:
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
Assertion (A): For any two prime numbers $p$ and $q$, their HCF is 1 and LCM is $p + q$.
Reason (R): For any two natural numbers, HCF × LCM = product of numbers.