Prove that a^3 ≡ a (mod 3) for every positive integer a.

Question

What I did:

Assume a^3 ≡ a (mod 3) is true for every positive integer a.
Then 3a^3 ≡ 3a (mod 3).
(3a^3 - 3a)/3 = k, where k is an integer
a^3 - a = k
Therefore, a^3 ≡ a (mod 3).

Is this a valid method for proving?

Answer 1

not really. Try this:

math.stackexchange.com/questions/992245/prove-that-if-a-in-mathbbz-then-a3-equiv-amod-3