1981 AHSME Problems/Problem 16

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 1

Since $9=3^2$ , every $2$ digits in base $3$ corresponds to $1$ digit in base $9$ . Since this number is $20$ digits long, which is an even number of digits, the answer must correspond to the first $2$ digits on the left. So the answer is $12_{3} = 1 \cdot 3 + 2 = 5\ \fbox{(E)}$ .

-edited by coolmath34, j314andrews

Solution 2 (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

1981 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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1981 AHSME Problems/Problem 16

Contents

Problem

Solution 1

Solution 2 (Long Way)

See also