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

Latest revision as of 19:48, 23 November 2021

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-==Problem==
+#REDIRECT [[2021_Fall_AMC_12A_Problems/Problem_10]]
-The base-nine representation of the number <math>N</math> is <math>27,006,000,052_{\text{nine}}</math>. What is the remainder when <math>N</math> is divided by <math>5</math>?
-<math>\textbf{(A)}\ 0  \qquad\textbf{(B)}\  1 \qquad\textbf{(C)}\  2 \qquad\textbf{(D)}\
-\qquad\textbf{(E)}\ 4</math>
-==Solution==
-Using module rules, we can find the remainder:
-<math>27,006,000,052_9 = 2(9^{10})+7(9^9)+6(9^6)+5(9^1)+2</math>
-<math>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)</math>
-<math>2({-}1^{10})+7({-}1^9)+6({-}1^6)+5({-}1^1)+2\equiv 2-7+6-5+2(\text{mod }5)</math>
-<math>2-7+6-5+2\equiv -2(\text{mod }5)</math>
-<math>-2\equiv 3(\text{mod }5)</math>
-Thus, the answer is <math>3</math> or <math>\boxed{(D)}</math>.
--Aidensharp