|
|
| (5 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| − | ==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==
| |
| − | 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
| |
