How to prove that a particular form of numbers achieve all remainders modulo $n$?

Question

(Note: Here numbers mean positive integers, all variables used are positive integers)

This query is regarding a particular question where I need to prove that a particular form of numbers ("particular form of numbers", for example squares, cubes, $4n+1$ etc.) can attain every remainder modulo $n$ for some positive integer $n$.

So my question is, what is the general process to approach this type of problem?

For example suppose we need to find the type of numbers $k$ for which the sequence $1^2, 2^2, 3^2 , \cdots $ contains all the remainders modulo $k$. (That is , the equation $r^2 = x \pmod k$ has a solution $r$ for all $x=1,2, \cdots k$).

So far, I have not been able to make any rigourous proof. So please give some general method to approach these type of question.

Answer 1

Partial progress : Here's the proof that $n$ power numbers for any even $n>1$ (i.e. squares, $4$ th powers etc.) can't achieve all remainders modulo $k$ for $k \gt 2$.

Define sets $A$ and $B$ as $A=\{ km, km+1, \cdots , km+k-1 \} $ $B=\{ 0,1,2 \cdots , k-1 \}$. See that $|A|=|B|$.

Define a map $p : A \mapsto B$ such that $p(a)=r$ where $ a^n \equiv r \pmod k$.

For any $a \in A$, Note that $a^n \equiv (k-a)^n \pmod k$. (Since $n$ is even, it is trivial by binomial theorem)

$\implies p $ is not a one one map and because $|A|=|B|$, $p$ can not be onto.