+321 What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.
What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.
4 digits base 3
smallest 1000 = 3^3 = 27
biggest 2222 = 2*27+2*9+2*3+2 = 80
3 digits base 6
smallest 100 = 36
biggest 1000-1 = 6^3-1 = 4095
So the intersection of these is 36 to 80 there are 80-36+1 = 45 of them
sum of numbers from 36 to 80 = (45/2)(72+44) = 45/2*116= 45*58 = 2610
average = 2610/45 =58
A more straight forward approach would be to say that of there are 45 numbers the median on is in the 46/2 = 23rd position. In this case the median and the mean will be the same.
36+23-1 = 58 Just as expected.
So the answer is 58 base 10
