If [tex]\frac{x}{y}[/tex] is an integer, which of the following statements must be true?
A. Both x and y are integers
B. x is an integer
C. Either x or y is negative
D. [tex]\frac{y}{x}[/tex] is an integer
E. [tex]x = ny[/tex] where n is an integer
Answer (2)
Only E) is true. If I were to arbitrarily pick x = 2.2 and y =1.1, then I could show that A), B), C), and D) to be false, given x/y = 2.2/1.1 = 2 (which is an integer). However for E), if we define x/y = n where n is an integer, then by rearranging terms, we can show x = n*y where n still equals an integer and our original equation is still valid.
The only statement that must be true is E: x = n y where n is an integer, as it directly follows from the fact that y x is an integer. Other statements, A, B, C, and D, are not necessarily true in all cases.
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