Counts and skips the numbers that have digits divisible by 3 (change in base method) 
A man counts numbers and skips the numbers that have digits divisible by 3. The first ten numbers are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the 100th number?

Quick googling for solution gives an interesting alternative method using change in base:

Its actually base 6 system, So $100$ in base 6 is $244$ in decimal.
  But as $3$ is not included in the system, 4 will actually be 3, 5 will be 4, so on, Hence $100th$ number is 255

Can someone explain this please?
I tried same method on similar question,
to find $100th$ number for the sequence that skips digits divisible by 2 or 5.
100 in base 4 is 1210 and reconverting it back yields 1310. Which is wrong.
Many thanks in advance,
Chris
 A: The system you describe isn't quite like base six, because no digit functions as a zero digit. (So in standard base six, there are only $5$ one-digit numerals representing nonzero numbers: $1, 2, 3, 4, 5$; but in the system suggested in the problem, there are $6$: namely $1,2,4,5,7,8$.)  This means that you cannot have $0$ things in a place, but you can have $6$ things in a place.  This is not an issue if in the standard base six representation, there are no zeros in the numeral; but if there are zeros, then things change.  
In this funny base six if we used the familiar numerals $1, 2, 3, 4, 5, 6$, the sequence of numerals would be $1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15, 16, 21, ..., 55, 56, 61, 62, 63, 64, 65, 66, 111, ...$.  The place values would be as usual for base six (ones, sixes, thirty-sixes,...).  So for instance, $66$ would correspond to $6$ sixes and $6$ units, for a value of forty-two.  
But also in your system, the digits are $1,2, 4,5,7,8$, corresponding respectively to $1,2,3,4,5,6$ in funny base six.
So to find the $100$-th numeral in funny base $6$, we can write $100$ in standard base six: $244_{\text{six}}$.  Since there are no zeros, this would be the same as funny base six with standard digits.  But for your adjusted digits, you would use $255$ according to the correspondence above.
In your second example, you are using the digits $1,3,7,9$, which suggests a funny base $4$ with adjusted digits.  To find the $100$th funny adjusted number in this sequence, we write one hundred in standard base four: $1210_{\text{four}}$.  But this time we have a zero digit, so we actually have to do work to convert to funny base four.  We'd like to take from the next place, and regroup down, which would give us $1204$, but we now have a new zero, so we have to regroup again, obtaining $1144_{\text{funny four}}$ (one sixty-four, one sixteen, four fours, four units).  Then we have to translate to the digits we're using ($1, 3, 7, 9$) to obtain our desired value $1199_{\text{funny adjusted four}}$.
