2021 AMC Fall 10B Problem 5

Express everything in base $2$. Since $8 = 2^3$,
$$n = 8^{2022} = (2^3)^{2022} = 2^{6066}.$$
Dividing by $4 = 2^2$ subtracts $2$ from the exponent:
$$\frac{n}{4} = 2^{6066 - 2} = 2^{6064}.$$
The choices are powers of $4$, and $2^{6064} = (2^2)^{3032} = 4^{3032}$.

The answer is $\boxed{\textbf{(E)}\ 4^{3032}}$.

All the numbers involved are powers of $2$, so the only tool needed is the exponent rules $(a^b)^c = a^{bc}$ and $a^b / a^c = a^{b-c}$. Reduce to the smallest common base, do the arithmetic on exponents, then convert to the base used by the answer choices.

• Dividing the exponent by 4 or subtracting 4 from it (as in 8^2018) instead of subtracting 2 from the base-2 exponent.
• Computing $6066/2 = 3033$ and then subtracting $2$ again to get $4^{3031}$; the subtraction of $2$ happens in base $2$, which is only $1$ in base $4$.