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

Try to prove that I, II and III DON'T have to be integers.
To this end, plug in the MINIMUM POSSIBLE VALUE for y.

450y = n³ implies that 450y is the cube of an integer.

When we prime-factorize the cube of an integer, we get 3 (or a multiple of 3) of every prime factor:
8 is the cube of an integer because 8 = 2³ = 2*2*2.
27 is the cube of an integer because 27 = 3³ = 3*3*3.

Thus, when we prime-factorize 450y, we need to get AT LEAST 3 of every prime factor.
Here's the prime-factorization of 450y:
450y = 2 * 3² * 5² * y

Since 450 provides only one 2, two 3's, and two 5's, and we need at least 3 of every prime factor, the missing prime factors must be provided by y.
Thus, y must provide at at least two more 2's, one more 3, and one more 5.
Thus, the MINIMUM possible value of y = 2² * 3 * 5.

Plug y = 2² * 3 * 5 into the answer choices:

I. y/(3 x 2² x 5)
(2² * 3 * 5)/(3 x 2² x 5) = 1.
The smallest possible value of y yields an integer.
Eliminate every answer choice that does not include I.
Eliminate A, C and D.

II. y/(3² x 2 x 5)
(2² * 3 * 5)/(3² x 2² x 5) = 1/3.
Not an integer.
Eliminate every remaining answer choice that includes II.
Eliminate E.

The correct answer is B.