Difference between revisions of "1981 AHSME Problems/Problem 16"

(Created page with "==Problem== The base three representation of <math>x</math> is <cmath>12112211122211112222</cmath> The first digit (on the left) of the base nine representation of <math>x</m...")
 
Line 5: Line 5:
  
 
<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==
 
 
Notice that the value of the leftmost <math>1</math> is equal to <math>1 \cdot 3^{20}.</math> Converting this to base 9, you would get <math>1 \cdot 9^{10}.</math> Therefore, the leftmost digit is equal to 1, and the answer is <math> \textbf{(A)}\ 1.</math>
 
  
 
==Solution (Long Way)==
 
==Solution (Long Way)==

Revision as of 15:44, 28 January 2021

Problem

The base three representation of $x$ is \[12112211122211112222\] The first digit (on the left) of the base nine representation of $x$ is

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution (Long Way)

Convert $x$ to base 10 then convert the result to base 9. \[12112211122211112222_{3} = 8847859\]

\[8847859 = 17574874_{9}\]

Therefore, the answer is $\textbf{(A)}\ 1.$

-edited by coolmath34