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

Latest revision as of 20:56, 5 May 2021

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

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

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

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 $\textbf{(E)}\ 5.$

@@ 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>
-==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)==
 Convert <math>x</math> to base 10 then convert the result to base 9.
-<cmath>12112211122211112222_{3} = 8847859</cmath>
+<cmath>12112211122211112222_{3} = 2150029898</cmath>
-<cmath>8847859 = 17574874_{9}</cmath>
+<cmath>2150029898 = 5484584488_{9}</cmath>
-Therefore, the answer is <math> \textbf{(A)}\ 1.</math>
+Therefore, the answer is <math> \textbf{(E)}\ 5.</math>
 -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 <math> \textbf{(E)}\ 5.</math>