Number Theory
If x is an integer greater than 1 and if y = x + 1/x, which of the following must be true?
I. y ≠ x
II. y is an integer
III. xy > x2
(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III
Show/Hide Answer
Answer: Choice D
Level of Difficulty E M H
The best way to figure this out is to examine the extremes. It says x is an integer greater than 1, so:
If we let x have its smallest value of 2, y will have a value of 2.5 .
If we let x approach infinity, y also approaches infinity, but is always slightly larger than x.
Therefore, as x takes integer values from 2 to infinity, y takes on values from 2.5 to infinity, and is always slightly larger than x.
Notice:
Case I is true, since x and y will never have the same value.
Case II is false since y will never be an integer.
Case III is true since if y is always bigger than x, then xy will always be larger than xx.
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