For a positive integer n, what is the remainder when n(n+1) is divided

ques- 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.

answer-

when N is divisible by 3, then N is a multiple of 3 (N can be 3,6,9,12...etc)
so this case can have both situations where N(N+1) is divisible by 12 and not divisible by 12.
For example- let N=3, thus 3*4 is divisible by 12, but if N=6 then 6*7 is not divisible by 12.
we are not certain what the remainder will be. It can either be Zero or any other number

this statement is NOT SUFFICIENT. Thus we can eliminate choices A and D. we are now left with choices B,C,E

following the same above approach for statement 2 when N is divisible by 4, we can have two situations where N(N+1) is divisible by 12 ( for N=8) but N(N+1) is not divisible by 12 (for N=4,16..etc). thus again we are not certain what will be the remainder. it can either be Zero or any other number

So this statement is also NOT SUFFICIENT. We can eliminate choice B. 2 choices are left-C,E

now if we are given N is divisible by both 3 and 4, then N is a multiple of 12. Thus, N(N+1) is divisible by 12. now we are certain about the remainder
ANswer is C

==> In the original condition, there is 1 variable (n), so D is highly likely to be the answer. In the case of 1), if =3, n(n+1)=12 with the remainder of 0,and if n=6, n(n+1)=42=12*3+6 with the remainder of 6, hence not sufficient. In the case of 2), if n=4, n(n+1)=20=12*1+8 with the remainder of 8, if n=8, n(n+1)=72=12*6 with the remainder of 0, hence not sufficient. Through 1) & 2), n=12, 24, 36… all have the remainder of 0, hence unique, and sufficient. C is the answer.
Answer: C

Asked: For a positive integer n, what is the remainder when n(n+1) is divided by 12?
1) n is divisible by 3.
n = 3k
n(n+1) = 3k(3k+1) = 9k^2 + 3k
The remainder when n(n+1) is divided by 12 = {0, 6}
NOT SUFFICIENT

2) n is divisible by 4.
n = 4k
n(n+1) = 4k(4k + 1) = 16k^2 + 4k
The remainder when n(n+1) is divided by 12 = 4k(k+1) = {8,0}
NOT SUFFICIENT

(1) + (2)
1) n is divisible by 3.
n = 3k
n(n+1) = 3k(3k+1) = 9k^2 + 3k
The remainder when n(n+1) is divided by 12 = {0, 6}
2) n is divisible by 4.
n = 4k
n(n+1) = 4k(4k + 1) = 16k^2 + 4k
The remainder when n(n+1) is divided by 12 = 4k(k+1) = {8,0}
Combining, we get
The remainder when n(n+1) is divided by 12 = 0
SUFFICIENT

IMO C

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.