Let $N$ be a positive integer. Both when it is written in base 5 and when it is written in base 9, it is a 3-digit number, but the order of the numerals is reversed. We are to represent $N$ in base 10 (decimal) and in base 4. Let $N$ be $abc$ in base 5 and $cba$ in base 9. Then we have $$\mathbf { A } \leq a \leq \mathbf { B } , \quad \mathbf { C } \leq b \leq \mathbf { D } , \quad \mathbf { E } \leqq c \leqq \mathbf { F } \text {. }$$ Since we also have $$N = \mathbf { G H } a + \mathbf { I } \quad b + c = \mathbf { J K } c + \mathbf { L } \quad b + a ,$$ we obtain $$b = \mathbf { M } a - \mathbf { N O } c .$$ The $a$, $b$ and $c$ satisfying (1) and (2) are $$a = \mathbf { P } , \quad b = \mathbf { Q } , \quad c = \mathbf { R } .$$ Thus $N$ expressed in base 10 is $\mathbf { S T U }$, and $N$ expressed in base 4 is $\mathbf { V W X Y }$.
Let $N$ be a positive integer. Both when it is written in base 5 and when it is written in base 9, it is a 3-digit number, but the order of the numerals is reversed. We are to represent $N$ in base 10 (decimal) and in base 4.
Let $N$ be $abc$ in base 5 and $cba$ in base 9. Then we have
$$\mathbf { A } \leq a \leq \mathbf { B } , \quad \mathbf { C } \leq b \leq \mathbf { D } , \quad \mathbf { E } \leqq c \leqq \mathbf { F } \text {. }$$
Since we also have
$$N = \mathbf { G H } a + \mathbf { I } \quad b + c = \mathbf { J K } c + \mathbf { L } \quad b + a ,$$
we obtain
$$b = \mathbf { M } a - \mathbf { N O } c .$$
The $a$, $b$ and $c$ satisfying (1) and (2) are
$$a = \mathbf { P } , \quad b = \mathbf { Q } , \quad c = \mathbf { R } .$$
Thus $N$ expressed in base 10 is $\mathbf { S T U }$, and $N$ expressed in base 4 is $\mathbf { V W X Y }$.