How can one calculate $342342^{1001}$ mod $5$? How can one calculate $342343^2$ mod $3$? I know that the answer is $1$.
And $342342^{1001}$ mod $5$. 
I know that
$
3^0 \mod 5 = 1 \\
3^1 \mod 5 = 3 \\
3^2 \mod 5 = 4 \\
3^3 \mod 5 = 2 \\\\
3^4 \mod 5 = 1 \\
3^5 \mod 5 = 3 \\
3^6 \mod 5 = 4 \\
$
So 1001 = 250 + 250 + 250 + 250 + 1, which is why the answer is also 1?

First, Note that 
$$342342 = 34234*10+2= 34234*2*5 +2$$
So you have 
$$342342 \equiv 2 \pmod 5$$
Then, remember that
$$\forall a,b,c,n \in \mathbb N, a \equiv b \pmod n \implies a^c \equiv b^c \pmod n$$
Therefore,
$$342342^{1001} \equiv 2^{1001} \pmod 5$$
Finaly, note that $1001 = 1000+1 = 4*250 +1$ and try to conclude
