i've been given this question as part of an assignment for a university maths subject.
"Prove that N squared minus 2 is never divisible by 3 if n is an integer"
any help anyone could give me on how to attack this proof would be great.
Thanks heaps in advance.
If N is divisible by 3 then N=3k, for some integer k, and N^2=9k^2,
so N^2-2=9k^2-2, which is not divisible by 3.
If N leaves remainder 1 when divided by 3 then N=3k+1, for some integer k,
and N^2=9k^2+6k+1, so N^2-2=9k^2+6k-1, which is not divisible by 3.
If N leaves remainder 2 when divided by 3 then N=3k+2, for some integer k,
and N^2=9k^2+12k+4, so N^2-2=9k^2+6k+2, which is not divisible by 3.
This exhausts all the possibilities of the remainder when N id divided by 3,
and in no case was N^2-2 divisible by 3, therefore for no N is N^2-2 divisible
by 3.
RonL