$$a_1 = 2012$$
$$a_2 = 2012^{a_1} = 2012^2012$$
$$a_3 = 2012^{a_2} = 2012^{2012^2012}$$
What is the ones digit of $$a_2012$$ ?
What is the tens digit?

Question

anonymous · Answer

is it to the power of?

anonymous · Answer

$$a_1 = 2012$$
$$a_2 = 2012^{a_1}=2012^{2012}$$
$$a_3 = 2012^{a_2}=2012^{(2012^{2012})}$$
Thanks!

anonymous · Answer

and what's the question?

anonymous · Answer

What are the last two digits of $$a_{2012}$$

anonymous · Answer

last digit is 6

anonymous · Answer

$$2012 \equiv 12 \mod 1000 \implies 2012^{{2012}^{2012...}} \equiv 12^{{2012}^{2012...}} $$

anonymous · Answer

Show that the last two digits of the 2012th Ackermann number $$ 2012 \uparrow \uparrow 2012$$ are 16.

anonymous · Answer

You need to use a technique known as Euler's tower. Basically you repeatedly use Euler's Theorem, giving you $$\displaystyle{2012^{\cdots}\mod(100) \equiv 2012^{2012^{\cdots}\mod(40)}\mod(100) \equiv \cdots}$$ Eventually, you'll just be taking 2012 to some large power mod 2 (after a few more Euler's Theorem applications), cutting the number of terms in your power tower from 2012 to 6 or 7 or so. Then it's just a straight up computation to show the final two digits are 16. I'd finish it off for you, but the numbers would be too small to read!!