A $2$-digit number not containing a $0$, is raised to its $5$ th power. The last $5$ digits of this power are given.
How can we determine the original number ?
The number can be uniquely determined because the residues modulo $10^5$ are all distinct. The second digit of the original number is trivial to find. It is just the last digit of the given digit-string.
But how can we determine the first digit without brute force ? It would be best if the digit could be determined without electronic help fast. Any ideas ?