Multiple mod powers $(3^{4^{101}} \pmod{24})$

Question

$(3^{4^{101}} \pmod{24})$

I'm very inclined to get $4^{101} \pmod{24}$.

But if I have $a \equiv b \pmod{c}$ then is it necessarily true that $f^a \equiv f^b \pmod{c}$? Which is arguably what I was planning to do.

Answer 1

One can easily verify the following congruence $$3^3=3\pmod{24},$$ so we have $$3^4=3^2\pmod{24}.$$ Next we can obtain recursively the desired result: $$ \begin{split} (3^4)^4 &= (3^2)^4=(3^4)^2=3^2\pmod{24}\\ 3^{4^3} &= (3^2)^4=3^2\pmod{24}\\ &\dots\\ 3^{4^{101}}&=(3^2)^4=3^2=9\pmod{24}. \end{split} $$