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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 5? Write your answer in base 10.

 Apr 20, 2022
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Hint: If a number \(N\) in base 10 has 4 digits when written in base 3, that means \(27 \leq N < 81\), because 27 is the least number to have 4 digits in base 3 (which is 1000), and 81 is the least number to have 5 digits in base 3 (which is 10000).

 

Also, if it has 3 digits when written in base 5, \(25 \leq N < 125\) for a similar reason.

 

If an integer satisfies both inequalities, that means \(27 \leq N < 81\).

 

I believe you can continue on your own.

But in case you are still clueless, feel free to tell me.

 Apr 20, 2022

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