Show that for all integers n>2, n does not divide n^2+2

[numba]

Homework Statement

Show that for all integers n>2, n does not divide n^2+2.

2. The attempt at a solution
I believe this solution can be solved by induction, I just don't know how to phrase it recursively.

For all n>2, n^2+2 mod n ≠ 0

Base case n=3
3^2 + 2 =11
11 mod 3 = 2 ≠ 0.

Show that
(n-1)^2 +2 mod n-1 ≠ 0

 

[Office_Shredder]

I don't think induction is the right way to do this. n divides n², can you use that in this problem?

 

[numba]

I can use that, but I'm not sure how it would be done. Intuitively it makes sense, but is there an axiom that I could use that proves this?

 

[HallsofIvy]

Suppose n does divide n²+ 2. Then there exist integer m such that mn= n²+ 2. Now divide both sides by n.