The least perfect square, which is divisible by each of $21$, $36$ and $66$, is

### Concept & Strategy Find the Least Common Multiple (LCM) of the given numbers, express it as a product of its prime factors, and multiply by any missing prime numbers required to make all exponents even. ### Step-by-Step Solution Find the prime factorizations for $21$, $36$, and $66$. $21 = 3^1 \times 7^1$ $36 = 2^2 \times 3^2$ $66 = 2^1 \times 3^1 \times 11^1$ Determine the LCM by taking the highest power of each unique prime factor: LCM $= 2^2 \times 3^2 \times 7^1 \times 11^1$ Calculate the numerical value of the LCM: LCM $= 4 \times 9 \times 7 \times 11 = 36 \times 77 = 2772$. Check the parity of the exponents in the LCM's prime factorization. $2^2$ (even) $3^2$ (even) $7^1$ (odd, needs another $7$) $11^1$ (odd, needs another $11$) To convert the LCM into a perfect square, multiply it by $7 \times 11$. Multiplier $= 77$. Calculate the final square: Least Perfect Square $= 2772 \times 77$. $2772 \times 77 = 213444$. ### Exam Strategy & Shortcut Use divisibility rules to eliminate options. The number must be divisible by $36$, meaning it must be divisible by $9$. The sum of the digits must be a multiple of $9$. Sum of (a) $213444 = 18$ (Divisible by 9). Sum of (b) $214344 = 18$ (Divisible by 9). Sum of (c) $214434 = 18$ (Divisible by 9). Sum of (d) $231444 = 18$ (Divisible by 9). Since all are divisible by 9, you must calculate the base multiplication $2772 \times 77$. To do this quickly: $(2772 \times 70) + (2772 \times 7) = 194040 + 19404 = 213444$. ### Common Pitfall Miscalculating the prime factorization of $36$ or $66$, leading to an incorrect LCM base. Always double-check your prime factor trees before multiplying out large numbers. ### Final Answer Therefore, the correct answer is **213444**.