I am not sure if this is a legit thing to do, so a second opinion would be nice .. also if u see any better/easier ways to do it, please let me know:

Prove that n^5 is congruent (will be denoted by =) to n mod 5

PMI

Basis: 1^5 = 1 = 1 mod 5 as needed

Inductive:

Assume k^5 = k mod n

k^5 + 1^5 = k mod 5 + 1^5 (this and the next step are the ones i question)

so k^5 + 1 ^5 = k mod 5 + 1 mod 5 (bc 1^5 is always = 1 mod 5)

So (k+1)^5 = (k+1) mod 5 as needed

is this ok?