If n and y are positive integers and 450y = n³, which of the following must be an integer?

I. y/(3 x 2² x 5)

II. y/(3² x 2 x 5)

III. y/(3 x 2 x 5²)

a. None
b. I only
c. II only
d. III only
e. I, II, and III

It almost always helps to find the prime factorization in these question types where we ask whether a certain rational expression is an integer.

450y = n³
(2)(3)(3)(5)(5)(y) = n³
For (2)(3)(3)(5)(5)(y) to be a cube, the NUMBER OF 2's, 3's and 5's in the prime factorization must be divisible by 3.
So, for example, (2)(2)(2)(2)(2)(2)(3)(3)(3)(5)(5)(5) = [(2)(2)(3)(5)]³
Here, (2)(2)(2)(2)(2)(2)(3)(3)(3)(5)(5)(5) has six 2's, three 3's and three 5's

For (2)(3)(3)(5)(5)(y) to be a cube, it must be the case that the prime factorization of y includes at least two additional 2's, one additional 3 and one additional 5.
So, y = (2)(2)(3)(5)(other possible numbers)

Now check the options.

I. Must y/(3 * 2² * 5) be an integer?
Plug in y to get: (2)(2)(3)(5)(other possible numbers)/(3 * 2² * 5)
= some integer
Since this MUST be an integer, we can ELIMINATE A, C and D, which leaves us with B or E.

II. Must y/(3² * 2 * 5) be an integer?
Plug in y to get: (2)(2)(3)(5)(other possible numbers)/(3² * 2 * 5)
= (2)(other possible numbers)/3
This is not necessarily an integer. We're looking for values that must be an integer
Since this need not be an integer, we can ELIMINATE E, which leaves us with B.

NOTE: At this point we have the correct answer. But let's check III for "fun"

III. Must y/(3 * 2 * 5²) be an integer?
Plug in y to get: (2)(2)(3)(5)(other possible numbers)/(3 * 2 * 5²)
= (2)(other possible numbers)/5
Not necessarily an integer

Answer: B

Cheers,
Brent