Data Sufficiency

I got this one correct but looking for an easier approach, Thanks in advance


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.

himu 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.

Hi GMATGuruNY,

I was wondering, there seem to be (recently) a lot of data sufficiency questions posted that have the answer:
Statement 1 = INSUFFICIENT
Statement 2 = INSUFFICIENT
Combined = SUFFICIENT

Does this mean that this is the most common correct answer? I mean, if we knew the statistical history of such questions, and we found that, say, 88% of them had this answer, might we save ourselves some time in the test by blindly choosing it, to give us more time on another question? Do you know such historical information? Would this be an unwise move?