Question

What last number of 18^1000?

Answer 1

The last digit will simply be equal to 8^1000, since the 10 will not have any effect on the units digit.  Take the first few powers of 8, examining only the last digit:
8^1=8
8^2=4
8^3=2
8^4=6
8^5=8
8^6=4

We see that it continues in this pattern.  Powers evenly divisible by 4, therefore, end in 6.  Since 1000 is divisible by 4, 8^1000 ends in 6, so 18^1000 also ends in 6.