Question

If the remainder when x is divided by 4 is 3, then the remainder when $(2020+x)^{2022}$ is divided by 8 is ________ .

Hint:

In modular arithmetic, when dealing with large powers, the first step is to reduce the base. Then, look for a pattern or cycle in the powers of the reduced base. The property $(a \cdot b) \pmod n = ((a \pmod n) \cdot (b \pmod n)) \pmod n$ is fundamental.

Answer 1

Correct Answer: 1

\[ x\equiv3\pmod4 \] \[ 2020\equiv4\pmod8 \Rightarrow 2020+x\equiv3\text{ or }7\pmod8 \] \[ 3^2\equiv7^2\equiv1\pmod8 \] \[ (2020+x)^{2022}\equiv1 \] \[ \boxed{1} \]