If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?	Choose the correct choice from the following choices:
A) 2
B) 3
C) 6
D) 12
E) 24

Rationale: 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 . answer : b .
Answer: B