Find an x which satisfies 13^x = 13 mod 5003^2.
I started by noting that the same x will solve 13^(x - 1) = 1 mod 5003^2.
Since 5003 is prime and of course does not divide 13, we know that 13^(5002) = 13^(5003 - 1) = 1 mod 5003.
I'm not sure how to move from mod 5003 to mod 5003^2, though i'm kind of guessing that the answer is x = 5002 for the 5003^2 case, but i'm not sure. does anyone know what to do or know of a useful theroem? Thanks.