Find the highest power of $4$ in $82! + 83!$ I'am only getting $4^{13}$ as answer, but the correct answer is $40$.
What am I missing?
We first find the maximum power of $2$ in $82!+83!=82!(1+83)=82!(84)$, since $83$ is odd we want the largest power of $2$ in $84!$.

Using the Legendre-Polignac formula this is:

$\sum\limits_{n=1}^{\infty}\lfloor\frac{84}{2^n}\rfloor=\lfloor\frac{84}{2}\rfloor+\lfloor\frac{84}{4}\rfloor+\lfloor\frac{84}{8}\rfloor+\lfloor\frac{84}{16}\rfloor+\lfloor\frac{84}{32}\rfloor+\lfloor\frac{84}{64}\rfloor=42+21+10+5+2+1=81$.

So the maximim power of $4$ that divides the number is $4^{40}$
$$82!+83!=82!\left(1+83\right)=82!\cdot 84$$
$$[\frac{82}{4}]+[\frac{82}{4^2}]+[\frac{82}{4^3}]=20+5+1=26$$
$$84=4\cdot21$$
Then $$4^{27}|(82!+83!)$$