Consider group $\Bbb Z_{pq}$. Consider the sets $\{[p],[2p],\cdots ,[(q-1)p]$ and $\{[q],[2q],\cdots ,[(p-1)q]$.
Show that there does not exist positive integers $r,s$ such that $r[a]=s[b]$ where $[a]\in \{[p],[2p],\cdots ,[(q-1)p]\}$ and $[b]\in \{[q],[2q],\cdots ,[(p-1)q]\}$.
Since $[a]\in \{[p],[2p],\cdots ,[(q-1)p]\}$ we have $[a]=[lp]$ where $1\le l\le q-1$ and $[b]=[kq]$ where $1\le k\le p-1$.
Assume that there exist positive integers $r,s$ such that $r[a]=s[b]\implies [rlp]=[skq]$
But I don't understand how to get a contradiction from above?
Will someone please help.