Difference between revisions of "1981 AHSME Problems/Problem 16"
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<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math> | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math> | ||
| − | ==Solution (Long Way)== | + | ==Solution 1== |
| + | Since <math>9=3^2</math>, every <math>2</math> digits in base <math>3</math> corresponds to <math>1</math> digit in base <math>9</math>. | ||
| + | Since this number is <math>20</math> digits long, which is an even number of digits, the answer must correspond to the first <math>2</math> digits on the left. So the answer is <math>12_{3} = 1 \cdot 3 + 2 = 5\ \fbox{(E)}</math>. | ||
| + | |||
| + | -edited by coolmath34, j314andrews | ||
| + | |||
| + | ==Solution 2 (Long Way)== | ||
Convert <math>x</math> to base 10 then convert the result to base 9. | Convert <math>x</math> to base 10 then convert the result to base 9. | ||
| − | <cmath>12112211122211112222_{3} = | + | <cmath>12112211122211112222_{3} = 2150029898</cmath> |
| − | <cmath> | + | <cmath>2150029898 = 5484584488_{9}</cmath> |
| − | Therefore, the answer is <math> \textbf{( | + | Therefore, the answer is <math> \textbf{(E)}\ 5.</math> |
-edited by coolmath34 | -edited by coolmath34 | ||
| + | |||
| + | ==See also== | ||
| + | |||
| + | {{AHSME box|year=1981|num-b=15|num-a=17}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:09, 28 June 2025
Problem
The base three representation of 
is
![\[12112211122211112222\]](http://latex.artofproblemsolving.com/1/9/e/19ee9a81d71f36177bfbd8f9886022b1ae264e14.png)
The first digit (on the left) of the base nine representation of is
Solution 1
Since 
, every 
digits in base 
corresponds to 
digit in base 
.
Since this number is 
digits long, which is an even number of digits, the answer must correspond to the first digits on the left. So the answer is 
.
-edited by coolmath34, j314andrews
Solution 2 (Long Way)
Convert to base 10 then convert the result to base 9.
![\[12112211122211112222_{3} = 2150029898\]](http://latex.artofproblemsolving.com/6/4/e/64ebb09efcb51d16c956f98d227a8502de88a8b0.png)
![\[2150029898 = 5484584488_{9}\]](http://latex.artofproblemsolving.com/9/d/e/9ded47e3858d6a4a0d1be36ac3ce6e923e00ac6f.png)
Therefore, the answer is 
-edited by coolmath34
See also
| 1981 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
