2021 Fall AMC 10A Problems/Problem 12

Problem

The base-nine representation of the number $N$ is $27,006,000,052_{\text{nine}}$ . What is the remainder when $N$ is divided by $5$ ?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Using module rules, we can find the remainder:

$27,006,000,052_9 = 2(9^{10})+7(9^9)+6(9^6)+5(9^1)+2$

$2(9^{10})+7(9^9)+6(9^6)+5(9^1)+2\equiv 2({-}1^{10})+7({-}1^9)+6({-}1^6)+5({-}1^1)+2 (\text{mod }5)$

$2({-}1^{10})+7({-}1^9)+6({-}1^6)+5({-}1^1)+2\equiv 2-7+6-5+2(\text{mod }5)$

$2-7+6-5+2\equiv -2(\text{mod }5)$

$-2\equiv 3(\text{mod }5)$

Thus, the answer is $\boxed{\textbf{(D)}\ 3}$ .

-Aidensharp