a

Short answer. otherwise there won't be part b. Given R, the function $P=P(R)$ and $Q=Q(R)$, as said, are fast. So we can use the algorithm of the following part, which is fast too.

Long answer. Given $R$, the candidates of the $P,Q$ pair population is only 1, so few to guess.

b

Given $N$. Starting guess $R \leq \sqrt{N}$ the largest integer, then following steps.

1. Get the $P=P(R)$ and $Q=Q(R)$. If $N=P Q$, then done.
2. Try $R \leftarrow P -1$, but remembering the $P$ already collected previously is the new $Q=R+1$, go to step 1 for searching a lower $P$ while there is still room for $P$.
3. Now the 'searching lower' fails so a similar 'searching high' starting from $R \geq \sqrt{N}$, and hope the asker was not lying about $N$ then there shall be an answer

In this naïve algorithm, the results from step 2 will have $P Q < N$ and the results from step 3 will have $P Q >N$ so they deem not the answer. So the correct answer shall be out in step 1 already.

Overall, it is still polynomial time of the size of the digits