<s>[INST] Show that $ a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $ Show that $  a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $
Show that$  a \equiv 1 \pmod{2^4 } \Rightarrow a^{2^{4-2}} \equiv 1 \pmod{2^4} $
Answer:
$ a \equiv 1 \pmod{2^3} \\ \Rightarrow  a^2 \equiv 1 \pmod{2^3} \\ \Rightarrow a^{2^{3-2}}=a^{2^1} \equiv 1 \pmod{2^3} $
Am I right?
 [/INST] That looks fine!
To generalise a bit, if $a \equiv1(\mathrm{mod}n)$, you see that $a^k \equiv 1(\mathrm{mod}n)$, and so $a^{k^l}\equiv 1(\mathrm{mod}n)$ whatever $k$ and $l$ are :)
</s>