Esquecinaldo has a terrible memory for remembering numbers, but excellent for remembering sequences of operations. Therefore, to remember his 5-digit bank code, he can remember that none of the digits are 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 it is the largest number that satisfies these conditions and has no repeated digits. What is this code?
The number starts with 25 because $5^{2}$ is the only power of 5 with 2 digits.

![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-29.jpg?height=60&width=320&top_left_y=2407&top_left_x=848)

The candidates for the last 2 digits are the powers of 2 with 2 digits: 16, 32, and 64:

$$
25 \_16, \quad 25 \_32, \quad 25 \_64 \text {. }
$$

The middle digit can be 3, 6, or 9. To choose among these numbers, remember that the sum of the 5 digits is odd, and since $2+5$ is odd, the sum of the last 3 digits must be even. In this case, we have the numbers

$$
25316, 25916, 25332, 25932, 25664
$$

Among these numbers, those that do not have repeated digits are

$$
25316, 25916 \text {. }
$$

Therefore, the code is 25916.