If a number is a perfect square, its prime factorization contains only even powers. For instance (2^2)(3^2) = 36 is a square, but (2^2)(3^3) = 108 is not.
The prime factorization of 5,400 is:
= 54(100)
= (9)(6)(10)(10)
= (3)(3)(3)(2)(2)(5)(2)(5)
= (2^3)(3^3)(5^2)
Since the powers of 2 and 3 are odd, we know that 5,400 is not a square. The factorization gives us a clue as to what the possible values of n could be. 5,400 times n must result in a prime factorization with all even exponents. To generate all even exponents, n must have at least one 2 and at least one 3:
(2^3)(3^3)(5^2) times (2)(3) = (2^4)(3^4)(5^2)
(2)(3) = 6. There's no way to generate all even exponents with a smaller value of n, so choice (D) is correct.