Remainder?

[kishugoel]

If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?
(1) 2 is not a factor of n.
(2) 3 is not a factor of n.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

[Makumajon]

(1) n is odd.

Say, n=1, 3, 5, 7, 9, 11, 13 ...
n^2-1=0, 8, 24, 48, 80, 120, 168
r=0, 8, 0, 0, 8, 0, 0. Insuff.

(2) Say n=1, 2, 4, 5, 7, 8, 10, 11
n^2-1=0, 3, 15, 24, 48, 63, 99, 120
r=0, 3, 15, 0, 0, 15, 3, 0. Insuff.

Combining n =1, 5, 7, 11, 13-->
r=0, 0, 0, 0. Suff.

[800Bob]

Neither statement alone is sufficient. With statement 1, (n - 1)(n + 1) can, for example, be 8 or 24, which yield two different remainders when divided by 24. With statement 2, (n - 1)(n + 1) can, for example, be 3 or 15, which yield two different remainders when divided by 24.

Considering the two statements together...

Statement 1 tells us that n is not even; therefore (n - 1) and (n + 1) are both even. And furthermore, because (n - 1) and (n + 1) are two consecutive even numbers, one of them is a multiple of 4. Therefore (n - 1)(n + 1) will be a multiple of 8.

Statement 2 tells us that n is not a multiple of 3; therefore either (n - 1) or (n + 1) must be a multiple of 3. Therefore (n - 1)(n + 1) will be a multiple of 3.

The two statements together, then, tell us that (n - 1)(n + 1) is a multiple of 8*3 = 24. So the remainder when divided by 24 will always be 0. Sufficient. The correct response is C.

[freak100]

I used the same method, but made a mistake when I took (n+1) as having a factor of 2 instead of a factor of 4. Thanks for the clarification.