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

Problem restatementDetermine the smallest square number that is a multiple of 21, 36, and 66.

Concept/ApproachThe least such square is the square of the least common multiple with all prime exponents adjusted to be even. Compute LCM, then multiply by missing prime factors to make exponents even.

Step-by-step calculation Prime factors: 21 = 3 × 7; 36 = 2² × 3²; 66 = 2 × 3 × 11 LCM = 2² × 3² × 7 × 11 = 4 × 9 × 77 = 2772 To be a perfect square, exponents must be even. Current odd primes: 7¹, 11¹. Multiply LCM by 7 × 11 = 77 ⇒ N = 2772 × 77 = 213444 Check: 213444 = (2 × 3 × 7 × 11)² = 462²

Verification/AlternativeConfirm divisibility: 462² is divisible by 21, 36, and 66 since 462 contains factors 2, 3, 7, 11 with sufficient multiplicity when squared.

Common pitfalls

• Taking just the LCM (2772) instead of making it a perfect square.
• Forgetting to square the odd-power primes.

Final Answer213444