I am asked to find the least residues of $5^1$, $5^2$, $5^3$, $5^4$, $5^5$, and $5^6$ modulo $7$ and show that they form a complete set of least residues modulo $7$ excluding $0$.
I found that the least residues are $5$, $4$, $6$, $2$, $3$ and $1$ respectively but I do not know how I am supposed to show that these form a complete set of least residues modulo $7$ excluding $0$?
On a related note, I am asked to repeat this computation for $2^1$, $2^2$, $2^2$, $2^3$, $2^4$, $2^4$, $2^5$, and $2^6$ modulo $7$. Again, I found the least residues to be $2$, $4$, $1$, $2$, $4$, $1$ but again I am not sure that how I should show that these numbers form a complete set of least residues modulo $7$ excluding $0$?
I know that the set of complete least residues modulo $7$ is $\{0,1,2,3,4,5,6\}$ but I am confused as what the question is asking and how I should show it.
There are only $7$ possible values for the residue, you have obtained every single one of them besides $0$, hence you have obtain the complete set excluding $0$.
$\{2,4,1\}$ does not include $3$, it is not a complete set.