2. Does there exist an integer $A$ such that each of the ten digits $0,1, \ldots, 9$ appears exactly once as a digit in exactly one of the numbers $A, A^{2}, A^{3}$.
2. Since the total number of digits in $A, A^{2}$ and $A^{3}$ is 10 , the total number digits in $32,32^{2}, 32^{3}$ is 11 and the total number of digits in $20,20^{2}, 20^{3}$ is 9 , any solution $A$ must satisfy $21 \leq A \leq 31$. Since the unit digits of $A, A^{2}, A^{3}$ are distinct, the unit digit of $A$ can only be $2,3,7,8$. Thus the only possible values of $A$ are $22,23,27,28$. None of them has the desired property. Thus no such number exists.