Vishnu88 wrote: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

Question is from KAPLAN Math workbook.

1) if 4 is in s , then it means, 16 and 64 are also in s, also 4 is not a source integer, then it means it must be either a square or cube of another integer, we know that sqrt(4)=2 or -2, as we are dealing with only positive numbers hence sqrt(4)=2, therefore 2 is also part of s, if 2 is there than 8 will also be there in s, therefore 1 alone is sufficient to answer the question.

2)64 is in s, than it means 64^2 and 64^3 will also be in s, also 64 is not a source integer, therefore either its square root will also be in s, i.e. sqrt(64)=8 can be in s or cuberoot(64)=4 can also be in s, hence 2 alone is not sufficient to answer the question, hence A

stat 1: 4 is in S and is not the source integer
therefore numbers in S can be 4 16 64 (8 can be in S) or 2 4 8 (8 cannot be in S) hence not sufficient

stat 2: 64 is in S and is not the source integer
therefore numbers can be 8 64 64^2 (8 cannot be in S) or 4 16 64 (8 can be in S) hence not sufficient

Together: S can have 4 16 64 plus the source integer which can be either 8 or 3,5,7... etc
hence both statements together are not sufficient.
IMO: E

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.