Data Sufficiency

If n is a positive integer, is n^2 - 1 divisible by 24?

(1) n is a prime number.
(2) n is greater than 191

OA C

guerrero wrote:If n is a positive integer, is n^2 - 1 divisible by 24?

(1) n is a prime number.
(2) n is greater than 191

OA C

Question rephrased: Is (n+1)(n-1) divisible by 24?

Statement 1: n is a prime number
If n=2, then (n+1)(n-1) = 3*1, which is NOT divisible by 24.
If n=23, then (n+1)(n-1) = 24*22, which IS divisible by 24.
INSUFFICIENT.

Statement 2: n is greater than 191
If n = 239, then (n+1)(n-1) = 240*238, which IS divisible by 24.
If n = 200, then (n+1)(n-1) = 201*199, which is ODD and thus NOT divisible by 24.
INSUFFICIENT.

Statements combined:
Rule 1: Of every 3 consecutive integers n-1, n, and n+1, exactly ONE will be a multiple of 3.
Since n is a prime number greater than 191, n must be an ODD integer that is NOT a multiple of 3.
Implication:
Either n-1 or n+1 MUST be a multiple of 3.

Rule 2: Of every 2 consecutive even integers, exactly one will be a multiple of 4, while the other will be an even integer that is not a multiple of 4.
Since n is odd, n-1 and n+1 are both EVEN.
Thus, either n-1 or n+1 is a multiple of 4, while the other is even but not a multiple of 4.
Implication:
(n-1)(n+1) must be a multiple of 4*2 = 8.

Result:
Since (n-1)(n+1) is a multiple of 3 and 8, it must be divisible by 24.
SUFFICIENT.

The correct answer is C.

Footnote to this one: the fact that p^2 - 1 is divisible by 24 for all primes p greater than 3 is something that elementary math test writers seem to love (that, and the fact that 1001 isn't prime, which comes up entirely too often), and it wouldn't surprise me at all to see this concept tested in some form on the GMAT.

Matt@VeritasPrep wrote:Footnote to this one: the fact that p^2 - 1 is divisible by 24 for all primes p greater than 3 is something that elementary math test writers seem to love (that, and the fact that 1001 isn't prime, which comes up entirely too often), and it wouldn't surprise me at all to see this concept tested in some form on the GMAT.

Sorry Matt I didnt understand the text in bold

faraz_jeddah wrote:

Matt@VeritasPrep wrote:Footnote to this one: the fact that p^2 - 1 is divisible by 24 for all primes p greater than 3 is something that elementary math test writers seem to love (that, and the fact that 1001 isn't prime, which comes up entirely too often), and it wouldn't surprise me at all to see this concept tested in some form on the GMAT.

Sorry Matt I didnt understand the text in bold

Ah, OK!

1001 isn't a prime number, as 1001 = 7 * 11 * 13

I've seen a number of questions that try to trick you into thinking that this number is prime - it seems to be a favorite of math test writers. Another popular seemingly prime number is 221, which is really 13 * 17.

Gracias!

faraz_jeddah wrote:Gracias!

De nothing! (as they say in SoCal)