Number Theory

$100^{100} = (10^2)^{100} = 10^{200}.$
Now we can check the remainder of the powers of 10 being divided by 18.

$10^2 \equiv 10 mod(18) \because 10^2-(18*5) = 10$

$10^3 \equiv 10 mod(18) \because 10^3-(18*55) = 10$

$10^4 \equiv 10 mod(18) \because 10^4-(18*555) = 10$

$10^5 \equiv 10 mod(18) \because 10^5-(18*5555) = 10$

$10^6 \equiv 10 mod(18) \because 10^6-(18*55555) = 10$

...

Therefore we can conclude that the remainder of $100^{100}$ divided by 18 is 10

You could also look at it this way:

You can see that this will always end with 10/18, i.e. there will always be a remainder of 10.