complete residue wording question

[lizzy2]

Do you think this is an okay answer? I mean, to turn in:

Here is the question: Show that if 5 !| n (5 does not divide n), then n^4 = 1(mod 5).

Here is the answer I wrote out but I'm not sure if its a good answer:

We build a complete residue system (mod 5), {0, 1, 2, 3 , 4). So:
0^4 = 0(mod 5)
1^4 = 1(mod 5)
2^4 = 1(mod 5)
3^4 = 1(mod 5)
4^4 = 1(mod 5)

Since 5 !| {1, 2, 3, 4} (since 5 does not divide any number in the set), n^4 = 1(mod 5)

Comments:
Is it okay to sort of exhaust the complete residue system in our answers? We have already covered "if a = b(mod n) and c = d(mod n), then ac = bd(mod n)."

 

[arthur ohlsten]

I don't know if it is correct to exhaust the residue system, but that is the approach I would take. Your answer is correct, but I am poor on rigour.
Perhaps someone else will answer
Arthur

 

[lizzy2]

thanks

thank you for the reply and I'm happy someone else is saying, "That's an ok answer" - its not like a normal proof but as i understand it, it is sufficient.