Track / Stage 4 / 127 of 340 #387 of 1964

Problem 387

AMC 12 late, AIME early

Number theory Difficulty 4.8 Multiple choice

Let $a=1+3^{1}+3^{2}+3^{3}+3^{4}+\ldots+3^{999}+3^{1000}$ , then the remainder when $a$ is divided by 4 is:

Pick one

Official solution

Answer: $\mathbf{C}$ .
Solution: Since $a=1+3(1+3)+3^{3}(1+3)+3^{5}(1+3)+\cdots+3^{999}(1+3)$ , the remainder when $a$ is divided by 4 is 1. Therefore, the correct choice is C.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.