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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...")
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Revision as of 15:03, 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

Notice that the value of the leftmost $1$ is equal to $1 \cdot 3^{20}.$ Converting this to base 9, you would get $1 \cdot 9^{10}.$ Therefore, the leftmost digit is equal to 1, and the answer is $\textbf{(A)}\ 1.$

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

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