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) } 3\qquad\textbf{(E) }4</math>
-==Solution==
-Recall that <math>9\equiv-1\pmod{5}.</math> We expand <math>N</math> by the definition of bases:
-<cmath>\begin{align*}
-N&=27{,}006{,}000{,}052_9 \\
-&= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\
-&\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 5\cdot(-1) + 2 &&\pmod{5} \\
-&= 2-7+6-5+2 \\
-&= -2 \\
-&\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}.
-\end{align*}</cmath>
--Aidensharp ~kante314 ~MRENTHUSIASM
-==See Also==
-{{AMC10 box|year=2021 Fall|ab=A|num-b=11|num-a=13}}
-{{MAA Notice}}