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

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

Page

Toolbox

Search

1981 AHSME Problems/Problem 16

Problem

Solution

Solution (Long Way)