What is the last digit of $2003^{2003}$? 
What is the last digit of this number?
$$2003^{2003}$$

Thanks in advance.
I don't have any kind of idea how to solve this. Except that $3^3$ is $27$.

When calculating the last digit of $a\cdot b$ you only need to find the product of the last digit of $a$ and the last digit of $b$.
so $2003^{2003}$ ends in the same digit as $3^{2003}$
$3^1$ ends in $3$
$3^2$ ends in $9$
$3^3$ ends in $7$
$3^4$ ends in $1$
$3^5$ ends in $3$
$3^6$ ends in $9$
$3^7$ ends in $7$
$3^8$ ends in $1$.
So $3^{4k}$ ends in $1$.
From here $3^{2000}$ ends in $1$ (since $2000$ is $4\cdot 500$).
Therefore $3^{2001}$ ends in $3$, $2^{2002}$ ends in $9$ and $3^{2003}$ ends in $7$
