Calculate $85^{2014}\bmod {82}$ Calculate the following equation $$85^{2014}\equiv x \bmod {82}$$
Answer is 73 in Wolphram Alpha, but I always lose myself doing the step by step

Key Idea $\ $ If the modulus $\, = a^n \pm 1,\,$ then exponents on $\,a\,$ can be reduced mod $\,n,\,$ e.g. here
$\qquad\ \ {\rm mod}\,\ \color{#c00}{a^4\!+\!1}:\,\  (\color{#c00}{a^4\!+\!1}\!+\!a)^{j+4k} \equiv\, a^{j+4k}\equiv\, a^j (\color{#c00}{a^4})^k \equiv\, a^j (\color{#c00}{-1})^k $
Yours is  $\ a = 3,\,\ j\!+\!4k = 2\! +\! 4\cdot 253 = 2014,\ $ so $\ 85^{2013}\equiv 3^2(-1)^{253}\equiv -9\pmod{82}$
