If n is a positive integer, is n^2 - n divisible by 12?

Official Solution:


If n is a positive integer, is \(n^2 - n\) divisible by 12?

(1) \(\frac{n}{11}\) is an integer

The above means that \(n\) is a multiple of 11.

If \(n=11\), then \(n^2 - n=n(n-1)=110=(not \ a \ multiple \ of \ 12);\)

If \(n=11*12\), then \(n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12).\)

Not sufficient.

(2) \(\frac{n}{19}\) is an integer

The above means that \(n\) is a multiple of 19.

If \(n=19\), then \(n^2 - n=n(n-1)=19*18=(not \ a \ multiple \ of \ 12);\)

If \(n=19*12\), then \(n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12).\)

Not sufficient.

(1)+(2) \(n\) is a multiple of both 11 and 19:

If \(n=11*19\), then \(n^2 - n=n(n-1)=11*19(11*19-1)=11*19*208=(not \ a \ multiple \ of \ 12);\)

If \(n=11*19*12\), then \(n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12).\)

Not sufficient.


Answer: E