If x and y are nonzero integers and 450x = 120y then which of the following must be an integer?
I. xy/60
II. 15x/4y
III. 4x/15y	Options.
A) I only
B) II only
C) I and III
D) I and II
E) I, II, and III
450x=120y reduce by 30 --&gt; 15x=4y --&gt; x is a multiple of 4 (y=15x/4 so x must be multiple of 4 for y to be an integer) and y is a multiple of 15 (x=4y/15y must be multiple of 15 for x to be an integer).   I. xy/60 --&gt; always true, as xy will be multiple of 4*15= 
<gadget id="calculator">4*15</gadget>
<output>60</output>
60;   II. 15x/4y --&gt; as 4y=15x then 15x/4y=15x/15x=115 /15= 
<gadget id="calculator">15 /15</gadget>
<output>1</output>
1=integer so this option is also always true;   III. 4x/15y --&gt; as x=4y/15 then 4x/15y=16/225≠integer, so this option is never true.   Answer: I and II (only).   ANSWER:I and II
<result>D</result>
If x and y are nonzero integers and 450x = 120y then which of the following must be an integer?
I. xy/60
II. 15x/4y
III. 4x/15y