Again, I have no clue how to start the problem
Find the last digit in 2003^2002 + 2001^2002
if you have a number which ends in three, what does the square of that number end in, i.e. a number ending in 3 can be written as
n = 10 a
1 + 3
so
n
2 = 100 a
12 + 60 a
1 + 9 = 10 a
2 + 9
So a number ending in 3 squared ends in 9. How about a number ending in 9 times a number ending in 3? Pretty soon you will start to repeat and you will have it ends in a certain value mod some number.
Now do the same thing for a number ending in 1.
For example, take a number ending in 2, we would have ending in 2, 4, 8, 6, 2, 4, 8, 6, 2, .... Notice the cycle is 4 long so if you raised the number ending in 2 to the power n>0, it would end in
2 if n=1 mod(4)
4 if n=2 mod(4)
8 if n=3 mod(4)
6 if n=0 mod(4)
Now a number ending in 4:
4 if n=1 mod(2)
6 if n=0 mod(2)
Suppose we have 2
7 + 4
6. 7=3 mod(4), so 2
7 ends in 8. 6=0 mod(2) so 4
6 ends in 6. If we add those numbers together we get a number ending in 4.