vinni.k wrote:If x is a positive integer, is x prime?

(1) x! is not divisible by 5
(2) x! is divisible by 6

OA is E

NOTE: I think this question should have some kind of text restricting the value of x to non-negative integers since the definition of factorials does not include negative values. For example, we don't know the value of (-3)!. So, I've reworded the question to look more GMAT-like.

Target question: Is x prime?

Statement 1: x! is not divisible by 5
Consider these two possible values of x:
Case a: x = 3 (since 3! = 6, and 6 is not divisible by 5), in which case x IS a prime integer
Case b: x = 4 (since 4! = 24, and 24 is not divisible by 5), in which case x is NOT a prime integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: statement 1 is essentially telling us that x < 5. Once x > 5, x! IS divisible by 5

Statement 2: x! is divisible by 6
Consider these two possible values of x:
Case a: x = 3 (since 3! = 6, and 6 is divisible by 6), in which case x IS a prime integer
Case b: x = 4 (since 4! = 24, and 24 is divisible by 6), in which case x is NOT a prime integer
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Aside: statement 2 is essentially telling us that x > 2. Once x > 3, x! is divisible by 6

Statements 1 and 2 combined
Consider these two possible values of x that satisfy both statements:
Case a: x = 3, in which case x IS a prime integer
Case b: x = 4, in which case x is NOT a prime integer
Since we still cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Aside: once we combine the statements, we know that 2 < x < 5. So, as you can see, x = 3 or x = 4

Cheers,
Brent