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

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==Problem==
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#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==
 
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>\boxed{\textbf{(D)}\ 3}</math>.
 
 
 
-Aidensharp
 
 
 
==Solution 2==
 
We convert this into base <math>10,</math> so
 
<cmath>2 \cdot 9^{10}+7 \cdot 9^9+6 \cdot 9^6+5 \cdot 9+2</cmath>
 
Notice that <math>9 \equiv -1 \mod 5,</math>
 
<cmath>2 \cdot (-1)^10+7 \cdot (-1)^9+6 \cdot (-1)^6+5 \cdot (-1)+2=2-7+6-5+2</cmath>
 
Simplifying, <math>-2 \mod 5 \implies 3 \mod 5.</math> So, the answer is <math>\boxed{3}.</math>
 
 
 
- kante314
 

Latest revision as of 19:48, 23 November 2021