modular arithmetic - 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?

Answer

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