S is a set of positive integers such that if integer X is a member of S, then both X2 and X3 are also in S. If the only member of S that is neither the square nor the cube of another member of S is called the source integer, is 8 in S?
1) 4 is in S and is not the source integer.
2) 64 is in S and is not the source integer.
Only the source integer is neither the square nor the cube of another member of S.
Thus, if X is not the source integer, then X is either the square or the cube of another member of S.
Statement 1: 4 is in S and is not the source integer.
Thus, 4 is either the square or the cube of another member of S.
Since 4 is not the cube of an integer, 4 must be the square of another member of S.
Thus, 2 is in S.
If 2 is in S, then 2²=4 and 2³=8 are in S.
Sufficient.
Statement 2: 64 is in S and is not the source integer.
Thus, 64 is either the square or the cube of another member of S.
If 64 is the square of another member of S, then 8 is in S.
If 64 is not the square but only the cube of another member of S, then we know that 4, 4²=16 and 4³=64 are all in S, but we cannot determine whether 8 is in S.
Insufficient.
The correct answer is
A.
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