Statement 1:

If N is not divisible by 2 then the numbers could be

0,1,2 (Remainder for (n-1)(n+1) = 0)
2,3,4 (Remainder for (n-1)(n+1) = 8)
4,5,6 (Remainder for (n-1)(n+1) = 0)
6,7,8 (Remainder for (n-1)(n+1) = 0)
8,9,10 (Remainder for (n-1)(n+1) = 8)

we see a trend that anytime N is divisible by 3, the remainder is 8. rest of the times, remainder is 0.

so 2 possibilities are 0 and 8,

Hence Insufficient

Statement 2:

If N is not divisible by 3 then the numbers could be

0,1,2 (Remainder for (n-1)(n+1) = 0)
1,2,3 (Remainder for (n-1)(n+1) = 3)
3,4,5 (Remainder for (n-1)(n+1) = 15)
4,5,6 (Remainder for (n-1)(n+1) = 0)
6,7,8 (Remainder for (n-1)(n+1) = 0)

we see a trend that anytime N is divisible by 2, the remainder is something other than 0. Rest of the times, remainder is 0

Hence, Insufficient

Combining, N is not divisible by both 2 and 3 - Hence, the remainder must always be 0. Thus C.

Also, it'll be great if somebody can suggest a quicker way of doing this - My method took me a long time.

Thanks

This may not help anybody but a guessing tip may be to always choose C when you see a problem like this. I still don't know how to do these problems but in each of the three problems the answer has been C. It may just be a coincidence but if you really don't know how to do it and you get it on the actual test, it may be something to keep in mind.