Dividing 100% by 3 without any left
So in mathematics, as far as I know, you can't divide 100% by 3, without having that 0,1..% left....
No! we can in Math and also in real life. This is similar to ask can we divide $1$ into 3 parts? And the answer is again yes. $$1÷3=\frac{1}{3}$$ because adding $\frac{1}{3}$ three times give $1$.
Consider 3 sticks of same length. Align these three sticks and call the total length as 1 unit. Now the length of any of the individual stick is exactly $\frac{1}{3}$ unit.
Moreover you can use the number system having base 3 to remove the (apparent) incompleteness of the base-ten expression $1÷3=0.333333333...$ In the number system having base 3, the number '3' itself will be written as $10$ and the number $1$ as it is.
The division $1÷3$ is now $1÷10$ which is equal to $0.1$. so you see writing (in base ten) $100÷3=33.333333$ does not mean that we cannot divide $100$ into three equal parts. What it means is that we are using a number system having $10$ base so we cannot write $\frac{100}{3}$ in decimals.
Meanwhile in ancient Greece...
For quite a long time, greek (and not only) mathematicians described numbers as lengths of certain line segments. So, when asked "What is $\sqrt{2}$ equal to?" they'd draw a $1\times1$ square (nevermind the unit), draw it's diagonal and say "There it is! This diagonal's length equals exactly $\sqrt{2}$!". So to answer your question: draw yourself a line, pick up a calliper, and divide this line 3 times. Like so:
And there you have it: 100% of a line segment divided into 3 equal parts. And if you ask "Yes, but what is this $\frac{1}{3}$ really equal to?" ancient philosopher would show you one of the parts and say "There it is! This segment's length equals exactly $\frac{1}{3}$!"
I think you have problems with this because you're thinking in base 10, and 10 (in base 10) doesn't divide evenly by 3. Think in base 3 instead. $100\%$ in base 3 is:
$100\% = 10\% + 10\% + 10\%$
Which are trivially demonstrated to be equal parts, with no remainder.
Addendum: although I have substituted its meaning analogously, $\%$ is the symbol for percent, which sets the base (10) squared of percents equal to the whole, 1, and the units of which increment by percentile. This is the concept of quantiles. In base three, the analogous term is the nonile (see definition of quantile in this Google book link, Discovering Statistics Using R), where the base, 3, squared, in units is set equal to the whole. This appears to be most commonly used in geometry and statistics, based on my own searches on this topic.
I am uncertain what the analogous term for percent itself would be, perhaps pernon?
Here the colored groupings of noniles illustrate the equation given above.