If \(n\) and \(y\) are positive integers and \(450y=n^3\),

BTGmoderatorLU wrote:Source: GMAT Prep

If \(n\) and \(y\) are positive integers and \(450y=n^3\), which of the following must be an integer?

I. \(\frac{y}{3\cdot 2^2 \cdot 5}\)

II. \(\frac{y}{3^2\cdot 2 \cdot 5}\)

III. \(\frac{y}{3\cdot 2 \cdot 5^2}\)

A. None
B. I only
C. II only
D. III only
E. I, II, and III

The OA is B

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

450y = n^3
2*3*3*5*5*y = n^3
For 2*3*3*5*5*y to be a cube, we need the number of 2's, 3's and 5's in the prime factorization to each be divisible by 3.
So, for example, 2*2*2*2*2*2*3*3*3*5*5*5 = (2*2*3*5)^3

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 option.

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

Answer: B

Cheers,
Brent

BTGmoderatorLU wrote:Source: GMAT Prep

If \(n\) and \(y\) are positive integers and \(450y=n^3\), which of the following must be an integer?

I. \(\frac{y}{3\cdot 2^2 \cdot 5}\)

II. \(\frac{y}{3^2\cdot 2 \cdot 5}\)

III. \(\frac{y}{3\cdot 2 \cdot 5^2}\)

A. None
B. I only
C. II only
D. III only
E. I, II, and III

The OA is B

Since 450 = 25 x 18 = 5^2 x 2 x 3^2, the smallest positive integer value for y is 5 x 2^2 x 3 since 450y is n^3, a perfect cube. We see that if y = 5 x 2^2 x 3, then only I is an integer.

Answer: B