<s>[INST] Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7 Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7
We have just started he section on modular arithmetic so I am new to a residue system, we did a similar problem that we as follows in class:
EX: m = 7
$-\frac{7 -1}{2}$ Which gave us a residue system of {- 3, -2 , -1, 0 , 1, 2, 3}
OR {0, 1, 2, 3, 4, 5, 6}
-Based off this example from class I am a bit confused as to how to approach this problem, any help is appreciated.  
 [/INST] \begin{array}{ccc}
-13 \equiv \color{red}{1} \mod 7 & -9 \equiv \color{red}5 \mod 7 & -4 \equiv \color{red}3 \mod 7  \\
-1 \equiv \color{red}6 \mod 7 & 9 \equiv \color{red}2 \mod 7 & 18 \equiv \color{red}4 \mod 7 \\
&21 \equiv \color{red}0 \mod 7&
\end{array}
Isn't this enough? Each member in your set is congruent to one of $\{0,1,2,3,4,5,6\}$ modulo $7$, and thus your set is a complete residue system.
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