I got 18^16 mod 25.
Since 18^4 = 1 mod 25, we have
18^16 = 1 mod 25.
I'm preparing for a contest, and for some of the practice questions, I've managed to simplify things greatly, but I can't figure out how to simplify what I have further.
For example:
17^17 mod 25
18^16 mod 25
19^19 mod 25
How do I simplify these expressions any further...
I don't think CRT and Euler's formula can help me any further.
A different approach is by the Binomial Theorem.
We have , namely , where is some integer. Then by the Binomial Theorem.
Note that all the terms except for the first one are divisible by 25. Therefore, . Hence, .
Similarly, from the congruence it follows that , so .
Again and then . Therefore ; so gives .