CIIM 2013 Problem 1

Given two natural numbers $m$ and $n$ , denote by $\overline{m.n}$ the number obtained by writing $m$ followed by $n$ after the decimal dot.

a) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has no solution.

b) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has a solution.