Simplifying $2^{30} \mod 3$ I simply cannot seem to get my head around this subject of mathematics. It seems so counter-intuitive to me. 
I have a question in my book: Simplify $2^{30}\mod 3$
This is my attempt: $2^{30}\mod 3 = (2\mod 3)^{30}$ (using the laws of congruences). 
$2\mod 3$ must be equal to $2$, right? $0\cdot 3 + 2 = 2$ then the rest is "2". Therefore, I concluded that $2^{30}$ is the answer. However, my book says $1$. How can this be? Can anyone explain this to me? It's probably really easy, and I feel kind of stupid for asking, but I simply don't get it.
And also, are there any sites on the internet that provide similar problems with which you can learn the concepts?
Thank you. 

You are being asked to find the least non-negative residue modulo $3$ (or perhaps the residue with least absolute value. For the first the options are $0,1,2$ and for the second you can choose $-1,0,1$. The residue is essentially the same as the remainder on division by $3$.
Now $2^{30}=(2^{10})^3=(1024)^3$ is of the order of $10^9$ and the point is to find a strategy to get a smaller value.
Now $1024\equiv 1$ is easy to establish. But the easiest route, once you have correctly identified the problem, is to notice that $2\equiv -1$. Quite often, these power problems involve finding a small power which is equivalent to $\pm 1$, since these are easy to work with.
For these kinds of problems - especially if they become more advanced - you should have Fermat's Little Theorem and the Euler-Fermat theorem to hand. But first get very used to the residue being the remainder on division.
