Question 169
If $A$ and $B$ are positive integers, then $\overline{AB}$ will denote the number obtained by writing, in order, the digits of $B$ after the digits of $A$. For instance, if $A = 193$ and $B = 2016$, then $\overline{AB} = 1932016$.
Prove that there are infinitely many perfect squares of the form $\overline{AB}$ in each of the following situations:
a) $A$ and $B$ are perfect squares;
b) $A$ and $B$ are perfect cubes;
c) $A$ is a perfect cube and $B$ is a perfect square;
d) $A$ is a perfect square and $B$ is a perfect cube.
Prove that there are infinitely many perfect squares of the form $\overline{AB}$ in each of the following situations:
a) $A$ and $B$ are perfect squares;
b) $A$ and $B$ are perfect cubes;
c) $A$ is a perfect cube and $B$ is a perfect square;
d) $A$ is a perfect square and $B$ is a perfect cube.