you missed my last part. I did give my OA as D. Just as a sidenote, we do not need to consider fractions as we are just considering integers. Integers are whole numbers.
the statement has not mentioned that x is an integer. therefore, we do not know x is an integer. if the question were like this, your reasoning would be correct:
Is X, an integer, to the power 3 an odd integer?
(1) 3 square root X is an odd integer.
(2) X is an odd integer.