I'm reading Niven, Zuckerman, Montgomery's number theory book and on page 51 they state this theorem:
Theorem 2.6: Let $(a,m)=1$. Let $r_1,r_2,\ldots, r_n$ be a complete, or a reduced, residue system modulo $m$. Then $ar_1,ar_2,\ldots,ar_n$ is a complete, or a reduced, residue system, respectively, modulo $m$.
Proof: If $(r_i,m)=1$, then $(ar_i,m)=1$ by theorem 1.8. There are the same number of $ar_1,ar_2,\ldots,ar_n$ as of $r_1,r_2,\ldots,r_n$. Therefore we need only show that $ar_i\not\equiv ar_j\pmod m$ if $i\neq j$. But thereom 2.3, part 2, shows that $ar_i\equiv ar_j \pmod m$ implies $r_i\equiv r_j \pmod m$ and hence $i=j$
I didn't understand why every integer is a residue of some of $ar_1,\ldots,ar_j$ modulo $m$.