x is a positive integer and (x − 1) is prime. Is x a prime number?

what if x = 5

then x-1 = 4--> not prime
x+2 + 7 -->Prime

Hi sun01,

x=5, fails to satisfy the condition given, i.e. x- 1 is a prime. Hence, x=5 is not the correct value to test.
Note: 2,3 are the only consecutive numbers that are prime numbers.

Thanks!

The question says
(x-1) is a prime (you wrote not prime)

I'd recommend at least starting with case testing for this one. Prime numbers aren't that predictable, so it's hard to make big general claims about them without looking at some numbers first.

Question stem: x - 1 is prime. Jot down some possibilities:

x-1 = 2, x = 3
x-1 = 3, x = 4
x-1 = 5, x = 6
x-1 = 7, x = 8
x-1 = 11, x = 12
x-1 = 13, x = 14
etc.
The question is asking, "is x somewhere on that list of numbers we just jotted down?"

Statement 1: x+2 is prime. Jot down some possibilities for the statement:

x+2 = 2: this fails, because x is a POSITIVE integer. Ignore this case.
x + 2 = 3, x = 1
x + 2 = 5, x = 3
x + 2 = 7, x = 5
x + 2 = 11, x = 9
x + 2 = 13, x = 11
etc.

The only possibility that shows up on BOTH lists is x = 3. (You can be sure that there aren't any other cases that work, because all of the rest of the numbers on this list will be odd, and all of the rest of the numbers on the list for the question stem will be even.)

Therefore, between this statement and the question stem, we know that x = 3. That's enough to answer the question, so this statement is sufficient.

Statement 2: Make another list.

x + 1 is not prime.

x + 1 = 1: this fails because x is a POSITIVE integer. Ignore this case.
x + 1 = 4, x = 3
x + 1 = 6, x = 5
x + 1 = 8, x = 7
x + 1 = 9, x = 8

Stop here! We've found two values of x that are on both lists: x = 3, and x = 8. One of them is prime, and the other one isn't. So, this statement is insufficient, because x is sometimes prime and sometimes not prime.

Therefore, only Statement 1 is sufficient, and the answer is A.

KAPLAN OFFICIAL EXPLANATION

Analyze the question stem

This Yes/No question indicates that x is a positive integer and that (x − 1) is prime, and it asks whether x is prime. Since prime numbers are mentioned, note that 2 is the only even and the smallest prime number. Thus, if (x − 1) is prime, then x is either 3 or an even number.

Evaluate the statements

Statement (1): If (x + 2) is prime, then x could not be an even number, since if x were even, then (x + 2) would also be even, and not 2, and thus could not be prime. Statement (1) can only be true if x = 3 (3 + 2 = 5, a prime), and so it is sufficient because the answer is always yes. Eliminate (B), (C), and (E).

Statement (2): If (x + 1) is not prime, then x could be 3. However, x could also be 8, 14, or any other even number that satisfies (x − 1) prime and (x + 1) not prime. Thus statement (2) is insufficient. (A) is correct.

TAKEAWAY: Understand the characteristics of prime numbers as well as the behaviors of odd and even numbers when added or subtracted.

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