If x and y are positive integers and 180x=y^3, which of the following must be an integer?
I. x/(2^2*3*5)
II. x/(2*3^2*5)
III. x/(2*3*5^2)	Choose one:
A) None
B) I only
C) II only
D) III only
E) I and II
x and y are integers and 180x=y^3 --&gt; 180x equals to cube of an integer (y^3).   180x=2^2∗3^2∗5∗x=y^3 ,The smallest value of x for which 2^2∗3∗5∗x is a cube of an integer is when x=2∗3∗5^2.to complete powers of 2, 3 and 5 to cubes).   In this case 180x=(2^2∗3^2∗5)∗(2∗3∗5^2)=(2∗3∗5)^3=cube of an integer So x is definitely divisible by 2∗3∗5^2   Answer: III only.
<result>D</result>
If x and y are positive integers and 180x=y^3, which of the following must be an integer?
I. x/(2^2*3*5)
II. x/(2*3^2*5)
III. x/(2*3*5^2)