How many different possible expressions are there for m^n if:

(i) both m and n are positive integers,not necessary distinct,

(ii) both numbers are below 10000 and

(iii) m^n can be written as k^6, k an integer?

Explain your result.

My car has a 5-digit odometer, which measures the miles since the car was built, and a 3-digit trip meter, which measures the miles since I last set it. Every so often, one or both of the readings is a palindrome. The meters reset to 000 after 999 and to 00000 after 99999.

The current readings are 123 and 12345. Assuming that I do not reset the trip meter, when is the next time both readings will be palindromes?

When was the most recent time both readings were palindromes?

Prove no matter what the mileage and trip meters read, they can eventually be made to both be palindromes without resetting the trip meter.

*Note: A palindrome reads the same forwards and backwards, like 262 or 37173.*