Is s an odd integer?

GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers. So, if there is no restriction then you must consider irrational numbers as well.

Note that only integers can be even or odd. So, for example there is no difference in asking "is s an odd integer" and "is s odd".

For more check Number Theory chapter of Math Book: https://gmatclub.com/forum/math-number-theory-88376.html

Back to the original question:

Is s an odd integer?

(1) \(\sqrt{s}\) is not an even integer --> \(\sqrt{s}\) might be an odd integer, for example 1, 3, ... and in this case \(s\) will be an odd integer, but \(\sqrt{s}\) might as well be a fraction, for example 1/3, 7/5 and in this case \(s\) won't be an integer at all. \(s\) also might be an irrational number, for example \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{{\frac{1}{3}}}\) ... and in this case \(s\) might or might not be an odd integer. Not sufficient.

(2) s^2 is not an even integer --> the same here: \(s^2\) might be a square of an odd integer, for example 1, 9, ... and in this case \(s\) will be an odd integer, but \(s^2\) might as well not be a perfect square, for example 1/3, 17, and in this case \(s\) won't be an integer at all. Not sufficient.

(1)+(2) When combined still insufficient: if \(s=1\) then the answer will be YES but if if \(s=\frac{1}{3}\) then the answer will be NO.

Answer: E.

Hope it's clear.