Difference between revisions of "2021 Fall AMC 10A Problems/Problem 12"

Revision as of 18:44, 22 November 2021

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

@@ Line 18: / Line 18: @@
 <math>-2\equiv 3(\text{mod }5)</math>
-Thus, the answer is <math>3</math> or <math>\boxed{(D)}</math>.
+Thus, the answer is <math>\boxed{\textbf{(D)}\ 3}</math>.
 -Aidensharp