Bunuel wrote:
If a and b are positive integers and \(90a = b^3\), which of the following must be an integer?
I. \(\frac{a}{2^2*3*5}\)
II. \(\frac{a}{2*3^2*5}\)
III. \(\frac{a}{2^2*3 *5^2}\)
(A) Only I
(B) Only II
(C) Only III
(D) Only I and II
(E) I, II and III
Now \(90a=b^3.......2*3^2*5*a=b^3\)
So a has to be a multiple of 2^2*3*5^2...
Thus a divided by a factor of 2^2*3*5^2 will be an integer..
I. \(\frac{a}{2^2*3*5}\)
2^2*3*5 is a factor of 2^2*3*5^2....so YES
II. \(\frac{a}{2*3^2*5}\)
2*3^2*5 is NOT a factor of 2^2*3*5^2....so Not necessary
III. \(\frac{a}{2^2*3 *5^2}\)
2^2*3*5^2 is a factor of 2^2*3*5^2....so YES
I and III
Bunuel, no choice matches the answer, so there must be a typo in choice I
You are right. Edited option I. Thank you.