Question

The least perfect square that is divisible by each of 321, 48 and 66 is:

• A. \(16 \times 1089 \times 11449\)
• B. \(1089 \times 9 \times 11449\)
• C. \(9 \times 121 \times 10404\)
• D. \(16 \times 1156 \times 11449\)

Hint:

To find the least perfect square multiple, find the LCM and multiply by any prime factors that have an odd power.

Answer 1

The Correct Option is A

Step 1: Prime Factorization:
\(321 = 3 \times 107\)
\(48 = 16 \times 3 = 2^4 \times 3\)
\(66 = 2 \times 3 \times 11\) Step 2: Find LCM and Make it a Perfect Square:
LCM must contain the highest powers: \(2^4, 3^1, 11^1, 107^1\). To become a perfect square, all exponents must be even. We need to multiply by \(3 \times 11 \times 107\). Required Square = \(2^4 \times 3^2 \times 11^2 \times 107^2\). Step 3: Match with Options:
\(2^4 = 16\). \(3^2 \times 11^2 = (33)^2 = 1089\). \(107^2 = 11449\). The number is \(16 \times 1089 \times 11449\).