1981 AHSME Problems/Problem 16

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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. \[12112211122211112222_{3} = 2150029898\]

\[2150029898 = 5484584488_{9}\]

Therefore, the answer is $\textbf{(E)}\ 5.$

-edited by coolmath34

Solution (Faster Way)

Every 2 numbers in base 3 represents 1 number in base 9. The first 2 numbers on the left,12 = 1(3) + 2(1) = 5.

So the answer is $\textbf{(E)}\ 5.$

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