a) Assume such integers exists. $p|b \implies b=pk$ and $q|a \implies a=ql$. Hence $ap+bq=1 \implies pq(k+l)=1$, which is not possible. So, there are no such pairs of integers. b) $p|a \implies a=pk$ and $q|b \implies b=ql$. Hence $ap+bq=1 \implies kp^2+lq^2=1$, which is possible since $\text{gcd}(p^2, q^2)=1$ c) From (b) suppose $a',b'$ are integers satisfying $a'p^2+b'q^2=1$. Then, $(a'p+q)p+(b'q-p)q=1$. Here $p$ does not divides $(a'p+q)$ and $q$ does not divides $(b'q-p)$. This proves the existence of such pairs.