Data Sufficiency (DS) | Re: If n is a positive integer and r is the remainder when (n - 1)(n + 1)

Algebraicapproach:

(1) n is not divisible by 2. Insufficient on its own, but this statement says that\(n=odd\) -->\(n-1\) and\(n+1\) are consecutive even integers -->\((n-1)(n+1)\) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...).

(2) n is not divisible by 3. Insufficient on its own, but form this statement either\(n-1\) or\(n+1\) must be divisible by 3 (as\(n-1\) ,\(n\) , and\(n+1\) are consecutive integers, so one of them must be divisible by 3, we are told that it's not\(n\) , hence either\(n-1\) or\(n+1\) ).

(1)+(2) From (1)\((n-1)(n+1)\) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by\(8*3=24\) , which means that remainder upon division\((n-1)(n+1)\) by 24 will be 0. Sufficient.

Answer: C.

Hope it'sclear.