Finding the remainder of a big power 
Find the remainder when $3^{29}$ is divided by $12$. 

Since $12=3•2^2$, this can be simplified to $3^{28}/4$. And the units digit of powers of 3 follows the pattern of $3,9,7,1$, so we know that $3^{28}$'s units digit is going to be $1$. However, that doesn't help much as $3^{28}$ divided by  4 can have a remainder of 1 or 3. How can I solve this without a calculator (I am not allowed to use one)? I feel like I could use a modulo, but since I'm not that familiar with it, I'm not sure. 

we have $$3^5\equiv 3 \mod 12$$ thus by squaring $$3^{10}\equiv 9\mod 12$$ again by squaring $$3^{20}\equiv 81\equiv 9\mod 12$$ and since $$3^9\equiv 3 \mod 12$$ we get
$$3^{29}\equiv 27\equiv 3 \mod 12$$
