You are given that $3^{400}\equiv 1\pmod{1000}$. What are the last three digits of $3^{12000}$?
I notice that $3^{12000}=(3^{400})^{30}$, so I can use the given congruence to simplify the expression modulo 1000. I have $3^{12000}\equiv (3^{400})^{30}\equiv 1^{30}\equiv 1\pmod{1000}$. 