swerve wrote:For a positive integer n, what is the remainder when n(n + 1) is divided by 12?

1) n is divisible by 3.
2) n is divisible by 4.

The OA is C.

I solve this question as follows,

12 = 4*3

We need to know if n is divisible by 4 and 3.

Statement 1: Insufficient. Since we don't know about n is divisible by 4.
Statement 2: Insufficient. Since we don't know about n is divisible by 3.

Both: Sufficient. n is divisible by 4 and 3, i.e n ( n+1) is divisible by 12. Hence C is the correct answer.

Given: n is a positive integer

To find out: The remainder when n(n + 1) is divided by 12.

Let's take each statement one by one.

1) n is divisible by 3.

Case 1: Say n = 3, thus n + 1 = 4 => n(n + 1) =3*4 = 12 is divided by 12. The answer is Yes.
Case 2: Say n = 9, thus n + 1 = 10 => n(n + 1) =9*10 = 90 is NOT divided by 12. The answer is No. No unique answer. Insufficient.

2) n is divisible by 4.

Case 1: Say n = 4, thus n + 1 = 5 => n(n + 1) =4*5 = 20 is NOT divided by 12. The answer is No.
Case 2: Say n = 8, thus n + 1 = 9 => n(n + 1) =8*9 = 72 is divided by 12. The answer is Yes. No unique answer. Insufficient.

(1) and (2) together

From (1) and (2), we know that n is divisible by 3*4 = 12. Since n is divisible by 12, n + 1 is divisible by 12. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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