1. The problem statement, all variables and given/known data Find the smallest natural number n such that: a^(3n)=a (mod 85) for each integer a. Justify your answer. 2. Relevant equations 3. The attempt at a solution because 85=17 . 5 and gcd (5,17)=1 we have to find the n such: a^(3n)=a (mod 5) and a^(3n)=a (mod 17). From Fermat theorem we know that a^(17)=a (mod 17) and a^(5)=a (mod 5) so we have: a^(5)=a^(3n) (mod 5) and a^(17)=a^(3n)(mod 17). I don't now how to continue and find the smallest natural n.