Calculate remainder modulo with double exponent.

Question

I recently stumbled upon this task:

$$ 10 * 2^{4+11^{85}} \equiv X \, mod \ 11 \\ Calculate \quad X$$

My approach was to split up and shorten the formula like this:

$$ 10*2^4 * (10* 2^{11^{85}}) = 16* 10 * (10* 2^{11^{85}}) $$

Now I don't know how to calculate the double exponent without a calculator .

My question is: how do I quickly calculate X by hand with the double exponent ?

Answer 1

From Fermat little theorem,

$$2^{10} \equiv 1 \mod 11$$

Let compute $$4+11^{85} \mod 10 \equiv 4+1 \mod 10 \equiv 5 \mod 10 $$

Hence the problem is equivalent to evaluate:

$$10 ( 2^5) \mod 11$$

You might like to note that $10 = 11-1$ and $32=33-1$.