Esquecinaldo has a terrible memory for remembering numbers, but excellent for remembering sequences of operations. Therefore, to remember his five-digit bank code, he can remember that the code does not have repeated digits, none of the digits is zero, the first two digits form a power of 5, the last two digits form a power of 2, the middle digit is a multiple of 3, and the sum of all the digits is an odd number. Now he no longer needs to memorize the number, because he knows that his code is the largest number that satisfies these conditions. What is this code?
The number starts with 25 because $5^{2}$ is the only power of 5 with two digits.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-096.jpg?height=65&width=342&top_left_y=887&top_left_x=937)

The candidates for the last two digits are the powers of 2 with two digits, namely, 16, 32, and 64. Since 32 is not suitable due to the repeated 2, we have the options

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-096.jpg?height=71&width=783&top_left_y=1141&top_left_x=722)

The middle digit is a multiple of 3, so it can only be 3, 6, or 9, but 6 cannot be repeated. To choose between the two options above, it is enough to remember that the sum of the five digits must be odd, and since $2+5$ is odd, the sum of the last three must be even. Thus, the second option above is discarded, as we cannot complete it with a multiple of 3, leaving only the numbers

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-096.jpg?height=66&width=777&top_left_y=1566&top_left_x=725)

The largest of the two, | 2 | 5 | 9 | 1 | 6 |
| :--- | :--- | :--- | :--- | :--- |
| , is the bank code of Esquecinaldo. |  |  |  |  |