(i) Find $8^{21} \pmod {23}$.
(ii) Solve the equation $8x ≡ 7 \pmod {23}$.
(i) Since 23 is a prime, and does not divide 8, we can use the FlT.
$ 8^{22} \equiv 1 \pmod {23} $
$ 8^{21} \times 8 \equiv 1 \pmod {23} $
By inspection we can see that $8^{21} \equiv 3 \pmod {23}$ gives us $3 \times 8 \equiv 24 \equiv 1 \pmod {23}$.
And so $8^{21} \equiv 3 \pmod {23}$.
(ii) From above we have $ 8^{21} \times 8 \equiv 1 \pmod {23} $ and $8^{21} \equiv 3 \pmod {23}$, and so
$ 3 \times 8 \equiv 1 \pmod {23} $
$ 3 \times 8 \times 7 \equiv 7 \pmod {23} $
Which gives us $x \equiv 21 \pmod {23}$.