M23-12

Official Solution:


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

(1) \(n\) is divisible by 11.

If \(n=11\), then \(n^2-n=n(n-1)=11*10\) and the answer is NO.

If \(n=11*12\), then \(n^2-n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12) =(a \ multiple \ of \ 12) \) and the answer is YES.

Not sufficient.

(2) \(n\) is divisible by 19.

If \(n=19\), then \(n^2-n=n(n-1)=19*18\) and the answer is NO.

If \(n=19*12\), then \(n^2-n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12) =(a \ multiple \ of \ 12) \) and the answer is YES.

Not sufficient.

(1)+(2) From the above, we have that \(n\) is divisible by \(11*19\):

If \(n=11*19\), then \(n^2-n=n(n-1)=11*19*(209-1)=11*19*208\) and the answer is NO (since \(11*19*208\) is not divisible by 3, it's not divisible by \(12=3*4\) either).

If \(n=11*19*12\), then \(n^2-n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12) =(a \ multiple \ of \ 12) \) and the answer is YES.

Not sufficient.


Answer: E