‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?

This is a moderately difficult problem that tests your knowledge on a specific remainder concept. This concept is,

"If ‘x’ is a prime number, (x-1)! will leave a remainder of (x-1) when divided by x. "

Put in simple words, this means that (x-1)! + 1 will be a multiple of x only if x is prime. Hence, in this question, we are essentially trying to find out if ‘x’ is prime.

From statement I alone, x can be 3 or 5 or 7 or 9. We cannot answer the main question with certainty, because 3,5, and 7 are prime while 9 is not. So, statement I alone is insufficient.
Answer options A and D can be ruled out, possible answer options are B, C or E.

From statement II alone, there cannot be any doubt at all. The statement says that ‘x’ is prime and this is what we are trying to find out.
In other words, we can say for sure that, since x is a prime numbers, (x-1)! + 1 will be divisible by x. Statement II alone is hence sufficient.

Therefore, the correct answer option is B.

If you were not familiar with the concept mentioned above, the alternative approach is to try values and see if a pattern develops. With statement I, you only have to try 4 values.
But, when it comes to statement II, it’s natural to worry about how many values you need to try, because there are infinitely many prime numbers. However, the fact about GMAT problems is, if a pattern is not there, it will become apparent to you in the first few instances itself; it’s not that the pattern breaker will reveal itself after 20 values. So, if you see a pattern developing after 4 or 5 values, go ahead and mark the answer.

Of course, the more foolproof of the two methods is the one using concepts. The best method, perhaps, is a combination of the both. Don’t you think so!

Hope this helps!

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