Find the last Digit of $237^{1002}$? I looked at alot of examples online and alot of videos on how to find the last digit  But the thing with their videos/examples was that the base wasn't a huge number. What I mean by that is you can actually do the calculations in your head. But let's say we are dealing with a $3$ digit base Number... then how would I find the last digit. 
Q: $237^{1002}$
EDIT: UNIVERSITY LEVEL QUESTION. 
It would be more appreciated if you can help answer in different ways. 
Since the Last digit is 7 --> 


*

*$7^1 = 7$

*$7^2 = 49 = 9$

*$7^3 = 343 = 3$ 

*$7^4 = 2401 = 1$ 
$.......$
$........$

*$7^9 = 40353607 = 7$

*$7^{10} = 282475249 = 9$


Notice the Pattern of the last digit. $7,9,3,1,7,9,3,1...$The last digit repeats in pattern that is 4 digits long.


*

*Remainder is 1 --> 7

*Remainder is 2 --> 9

*Remainder is 3 --> 3

*Remainder is 0 --> 1


So, $237/4 = 59$ with the remainder of $1$ which refers to $7$. So the last digit has to be $7$.

Finding the last digit of ${237}^{1002}$ is same as finding the last digit of $ 7^{1002}$.
We see that
$7^1 = 7$;
$7^2 = 9$;
$7^3 = 3$;
$7^4 = 1$;
$\dots$
Then comes the repetition of $7$, $9$, $3$ and $1$ as the powers of $7$ increase.
Since our power is even, our choices reduce to last digit of $7^2$ and $7^4$ which are $9$ and $1$ respectively.
Since $1002$ is a multiple of $2$ but not a multiple of $4$,
our choice of power $1002$ of $7$ is same as power $2$ of $7$, which is $9$.
Therefore, $7^{1002} = 9$ (last digit)
