M7MBA wrote:If n is a positive integer greater than 6, what is the remainder when n is divided by 6?

(1) n^2 - 1 is not divisible by 3.

(2) n^2 - 1 is even.

Approach 1:

Since n is greater than 6, and the statements refer to n², make a list of perfect squares greater than 6²:
n² = 49, 64, 81, 100, 121, 144, 169, 196, 225.
Subtracting 1 from these values, we get the following options for n² - 1:
n² - 1 = 48, 63, 80, 99, 120, 143, 168, 195, 224.

Statement 1: n² - 1 is not divisible by 3
From the list in red, the following options are viable:
80, 143, 224.
If n² - 1 = 80, then n=9, which yields a remainder of 3 when divided by 6.
If n² - 1 = 143, then n=12, which yields a remainder of 0 when divided by 6.
Since the remainder can be different values, INSUFFICIENT.

Statement 2: n² - 1 is even
From the list in red, the following options are viable:
48, 80, 120, 168, 224.
If n² - 1 = 48, then n=7, which yields a remainder of 1 when divided by 6.
If n² - 1 = 80, then n=9, which yields a remainder of 3 when divided by 6.
Since the remainder can be different values, INSUFFICIENT.

Statements combined:
From the list in red, the following options satisfy both statements:
80, 224.
If n² - 1 = 80, then n=9, which yields a remainder of 3 when divided by 6.
If n² - 1 = 224, then n=15, which yields a remainder of 3 when divided by 6.
Since the remainder is the same in each case, SUFFICIENT.

The correct answer is C.

Approach 2:

n-1, n, and n+1 are 3 consecutive integers.
Of every 3 consecutive integers, EXACTLY ONE must be a multiple of 3.

Statement 1: n² - 1 is not divisible by 3
(n-1)(n+1) = non-multiple of 3.
Implication:
Since neither n-1 nor n+1 is a multiple of 3 -- and one of every 3 consecutive integers must be a multiple of 3 -- n MUST BE A MULTIPLE OF 3.
If n=9, then dividing by 6 will yield a remainder of 3.
If n=12, then divided by 6 will yield a remainder of 0.
Since the remainder can be different values, INSUFFICIENT.

Statement 2: n² - 1 is even
(n-1)(n+1) = even.
Implication:
n-1 and n+1 must both be even, implying that n is ODD.
If n=7, then dividing by 6 will yield a remainder of 1.
If n=9, then dividing by 6 will yield a remainder of 3.
Since the remainder can be different values, INSUFFICIENT.

Statements combined:
Statement 1: n is a multiple of 3
Statement 2: n is odd
Options for n:
9, 15, 21, 27...
In every case, dividing by 6 will yield a remainder of 3.
SUFFICIENT.

The correct answer is C.