Question 47
For two positive integers $a$ and $b$ the number $\overline{a.b}$ is equal to the decimal fraction which we have if after the number $a$ we put the decimal point and then write the number $b$. For example, for $a = 20, b = 13$ we get $\overline{a.b} = 20.13$, and $\overline{b.a} = 13.2$.
Prove that there are infinite number of natural $n$, such that the equation $\overline{a, b} \cdot \overline{b, a} = n$ has no positive integer roots $a$ and $b$.
Prove that there are infinite number of natural $n$, such that the equation $\overline{a, b} \cdot \overline{b, a} = n$ has no positive integer roots $a$ and $b$.