Odd primes proof 
For distinct odd primes p and q, define the set
$S := ${$q · a − p · a': a, a' ∈ \mathbb{N} odd, a < p, a' < q$}
Prove that the numbers generated in S are (i)even, (ii)non-zero, and (iii)distinct

This is what I have so far
(i)The numbers generated in S are even. Since, q and p both are odd primes. $qa$ will be odd and $pa'$ will be odd. odd- odd = even
(ii) Since a and a' are less than p and q respectively and we know p and q are distinct, I can see through examples why $q · a − p · a'$ will be non-zero. However, I dont know how to put this in a general form.
(iii) Not sure how to show they'll be distinct either

(ii) If $qa-pa'=0$ we have $q/p=a'/a$, and since $q,p$ are primes thus relatively prime this implies $q/p$ is the most simplified form of this fraction. Hence $a'\geq q$ and $a\geq p$. It follows that for $a'<q,a<p$ the expression must be non-zero.
(iii) If $qa-pa'=qb-pb'$ for some $(a,b)\neq(a',b')$ we have $q(a-b)-p(a'-b')=0$ and thus by the argument used in (ii) we see that $q/p=(a-b)/(a'-b')$ which implies $a-b\geq q$ and $a'-b'\geq p$. But that is impossible if $a,b<q$ and $a',b'<p$.
