A well known demonstration for number-theoretical properties is the following :
A two-digit-number is raised to the $5$ th power and the original number has to be found.
It can be shown with the Carmichael function or with easy modular arithmetic, that the ending digit of the $5$ power is the same as the ending digit of the original number. We have
$$n^5\equiv n\mod 10$$ for every integer $n$
To get the first digit, one method is to cancel the last $5$ digits and to find $k$ , such that $k^5$ is smaller , but $(k+1)^5$ is larger than the resulting number (In the case of the ending digit $0$, the last non-zero digit is the first digit of the original number).
I wonder whether the first digit can be determined faster and without memorizing the $5$-th powers of the numbers $1$ to $9$.
For example, does determining the given number modulo $9$ or modulo $11$ help ?
What is the easiest way to find the first digit ?