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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>
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| − | <math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4</math>
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| − | ==Solution==
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| − | Recall that <math>9\equiv-1\pmod{5}.</math> We expand <math>N</math> by the definition of bases:
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| − | <cmath>\begin{align*}
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| − | N&=27{,}006{,}000{,}052_9 \\
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| − | &= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\
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| − | &\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 5\cdot(-1) + 2 &&\pmod{5} \\
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| − | &= 2-7+6-5+2 \\
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| − | &= -2 \\
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| − | &\equiv \boxed{\textbf{(D) } 3} &&\pmod{5}.
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| − | \end{align*}</cmath>
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| − | ~Aidensharp ~kante314 ~MRENTHUSIASM
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| − | ==See Also==
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| − | {{AMC10 box|year=2021 Fall|ab=A|num-b=11|num-a=13}}
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| − | {{MAA Notice}}
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