Q: If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?
Options:
(A) 2
(B) 3
(C) 6
(D) 12
(E) 24 Let's give some random thoughts before answering.
The prime factorization of a square has to have even powers of all its prime factors. If the original number has a factor, say of 7, then when it's squared, the square will have a factor of 7^2. Another way to say that is: any positive integer all of whose prime factors have even powers must be a perfect square of some other integer. Look at the prime factorization of 432
432 = (2^4)*(3^3)
The factor of 2 already has an even power - that's all set. The factor of 3 currently has an odd power. If n = 3, then 432*n would have an even power of 2 and an even power of 3; therefore, it would be a perfect square. Thus, n = 3 is a choice that makes 432*n a perfect square.
Therefore, the answer is (B).