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What is the average of all positive integers that have four digits when written in base 3, but three digits when written in base 6? Write your answer in base 10.

 
 Oct 14, 2021
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What is the average of all positive integers that have four digits when written in base 3, but three digits when written in base 6? Write your answer in base 10.

 

\(3^3\le x \le3^4-1\qquad \cup \qquad 6^2\le x\le6^3-1\\~\\ 27\le x \le 80 \qquad \cup \qquad 36 \le x\le 215\\~\\ 36\le x \le 80\)

 


\((36+37+ ....+80)  /  (80-36+1)\\~\\ =(36+37+ ....+80)  /  45\\~\\ =\frac{45}{2}(36+80) / 45\\~\\ =\frac{1}{2}(116) \\~\\ =58\)

 

 

Or I could have just said  36+  (80-36) / 2 = 36+22 = 58

 
 Oct 14, 2021

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