Number Theory: Complete set of residues modulo $n$
Solution 1:
Suppose to get a contradiction that $aa_i-aa_j=a(a_i-a_j)\equiv0\pmod n$ for some $i,j\in\{1,2\ldots,n\},$ $i\neq j.$ Then, since $\gcd(a,n)=1,$ $a_i-a_j\equiv0\pmod n$ (note that this is a consequence of the elementary Euclid's lemma, which states that if $\gcd(x,y)=1$ and $x\mid yz$ then $x\mid z$), which is clearly false.