The Problems of Ukrainian Authors

Let's designate through $P(n)$ the product of digits of the integer non-negative number $n$. Prove that sets $A$ and $B$ are unbounded, where:

a. $A=\left\{\frac{P(n)}{P(n^2)}\right\}$, where $n$ belongs to the set of such whole non-negative numbers that the number $n^2$ does not contain zero in the decimal record;

b. $B=\left\{\frac{P(n^2)}{P(n)}\right\}$, where $n$ belongs to the set of such whole non-negative numbers that the number $n$ does not contain zero in the decimal record.