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

Revision as of 20:52, 22 November 2021

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 $5?$

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

Solution

Recall that $9\equiv-1\pmod{5}.$ We expand $N$ by the definition of bases: $\begin{align*} N&=27{,}006{,}000{,}052_9 \\ &= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\ &\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 5\cdot(-1) + 2 &&\pmod{5} \\ &= 2-7+6-5+2 \\ &= -2 \\ &\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}. \end{align*}$ ~Aidensharp ~kante314 ~MRENTHUSIASM

2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 14: / Line 14: @@
 &\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}.
 \end{align*}</cmath>
--Aidensharp ~kante314 ~MRENTHUSIASM
+~Aidensharp ~kante314 ~MRENTHUSIASM
 ==See Also==
 {{AMC10 box|year=2021 Fall|ab=A|num-b=11|num-a=13}}
 {{MAA Notice}}

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

Revision as of 20:52, 22 November 2021

Problem

Solution

See Also