Solve $[2^{(2^{403})}] = [a]$ in $\mathbb Z_{23}$ where $0 \le a < 23$.

Question

Solve $[2^{(2^{403})}] = [a]$ in $\mathbb Z_{23}$ where $0 \le a < 23$.

I've tried to combine corollaries of Fermat's little theorem and various other methods to solve this, but have always been stuck.

Answer 1

Well, the order of $2$ modulo $23$ is $11$ (you can verify this). So, look at $2^{403}$ modulo $11.$

You will find the order of $2$ modulo $11$ to be $10$. This implies $$2^{403} \equiv 2^{10 \cdot 40+3} \equiv 2^3 \equiv 8 \text{mod}(23).$$

Thus, $$2^{2^{403}} \equiv 2^8 \equiv (2^5)(2^2)2 \equiv (9 \cdot 4) \cdot 2 \equiv 13 \cdot 2 \equiv 26 \equiv 3 \text{mod}(23).$$

Answer 2

Here's a way you could do it without orders of modulo...just in case you haven't learned it yet ;)

You can look at 2^403 modulo 22 (Splitting the modulus will make life easier). You'll get some equation involving 22 and you'll be able to use Fermat's little theorem.

Edit: on my phone, not sure how to format numbers