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What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 5? Write your answer in base 10.

 Dec 15, 2020
 #1
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I don't think there are any 4 digit3 that translate to 2 digit5

Smallest 4 digit base 3 is 1000  which is 2710

Largest base 5   two digit is 445 = 2410

 Dec 15, 2020
 #2
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Guest #1 is exactly right! Based on that, there 54 numbers from 27 to 80 inclusive that have 4 digits in base 3 and 3 digits in base 5. Their sum in base 10 =2,889. So, the average is: 2,889 / 54 =53.50 in base 10.

 Dec 15, 2020

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