my approach by picking numbers took 3.20 sec

(1) n is odd. This means that n-1 and n+1 are both even, so the product (n-1)(n+1) will be a product of 2 consecutive evens (one of which must be divisible by 4 and the other divisible by 2), so the product (n-1)(n+1) must be divisible by 8. If the product is 2*4=8, the remainder of division (8/24) will be 8. On the other hand if the product is 6*8=48, the remainder of division (48/24) will be 0. We don't know the remainder. INSUFFICIENT.

(2) This leaves too many numbers to be sufficient. If n=4, then the product (n-1)(n+1) will be 15 and remainder will be 15. On the other hand if n = 2, the product will be 3 and the remainder will be 3. INSUFFICIENT

The 2nd statement actually tells us that either n-1 or n+1 must be divisible by 3. This is because every 3rd integer is divisible by 3, so if it's not n, it must be n-1 or n+1.

Remember that statement 1 guaranteed that the product (n-1)(n+1) be divisible by 8.

Merging the statements tells us that the product is divisible by 3 and by 8, so it must be divisible by 24. The answer is C.

For a longer discussion or video solution, look at GMATPrep question 1141 or 1217

Good luck,
-Patrick