Determine the positive integer values of x less than 100 such that the last two digits of 2^x equal x.

What positive integer values of y less than 100 are there such that the last two digits of 3^y equal y?

There is exactly one result for each, according to the program I ran.

The program in fact finds that there is exactly one integer x in [1,99]  such  that (N^x)%100=x for  each integer N in [2,9].

As yet I have no idea how to find the solutions analytically, but presumably there is some  way to do that.

Posted by Richard on 2006-08-08 13:44:13