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What is the average of all positive integers that have three digits when written in base 5, but two digits when written in base 8? Write your answer in base 10.

 Jun 23, 2018
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OK, young person! Here is my attempt at this:

Here are ALL 3-digit numbers in base 5, which are 2-digit numbers in base 8:

 

100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144. If you count them you should get 25 such numbers. And their average is: [100 + 144] /2 = 122 in base 5, which is the exact median of the above list. Converting [100 + 144] / 2 =122  from base 5 to base 10, you get: [25 + 49] / 2 =37 in base 10.

 Jun 23, 2018

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