Solving problem on complete set of residue classes

Question

Let $m$ be a positive integer. Let $a$ be an integer relatively prime to $m$. $S$ is a reduced complete set of residue classes modulo $m$. Set $$T =aS=\{as | s ∈ S\},$$ then prove that $T$ is also a reduced complete set of residue classes modulo $n$.

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How to prove this? Will the $n$ be always less than $m$ as $aS$ will have only $m$ elements?

Answer 1

Hint: If $as_1 \equiv as_2 \pmod m$, then we have $m|a(s_1 -s_2)$. Now you need to show that $m \nmid a$ so $m|(s_1 - s_2)$. Now what can you conclude from $0 \le |s_1 - s_2| < m$ ?

Then you should have that $aS$ has $m$ distinct elements $\pmod m$ which means...

Answer 2

$T$ has at most $n$ elements $as_1$, $as_2$,...,$as_m$ (which, in principle may coincide modulo $m$). Let us suppose, for the sake of contradiction, that $$ as_i\equiv as_j\pmod{m} $$ for some $1\le i<j\le m$. Then $m$ divides $as_i-as_j=a(s_i-s_j)$. Since $a$ is coprime with $m$ then $m$ divides $s_i-s_j$, which is impossible since $S$ is a complete residue system modulo $m$.