The $2013$th digit of $1234567891011213141516\ldots$

How do I find the $2013$th digit of the string $12345678910111213141516\ldots$ I still don't get it, how are you suppose to find the exact digit. How did you hint help at all?

Hint: $\underbrace{123456789}_{9\text{ digits}}\underbrace{101112\dots979899}_{180\text{ digits}}\underbrace{100101102103\dots}_{2013-189\text{ digits}}$

There are 9 digits from 1 to 9. For the next ones, you have 10 11 12 13 14 15... Notice that the only difference is that there is a "1" preceding each of the digits. Therefore, that makes $9(2)$, or $18$ digits for the 10 - 19. Since we are going from $10$ to $99$, we know that there will be 10 times the amount of digits as there was from $10$ to $19$. That makes $18(10) = 180$ digits.

Counting the first 9, we have found $189$ digits so far. We have $2013-189 = 1824$ more digits until we reach digit number $2013$. These digits come in groups of three, i.e.:

100 101 102 103 104...

By dividing $1824$ by three, we can see how many pairs of three we need to count. The remainder will also tell us which digit in the pair is the correct digit. If we have a remainder of 0, we have the third digit of a three digit pair. If it is two, we must have the second digit in a three digit pair, and so on.

$\frac{1824}{3}$ has a remainder of $0$, so we know that we must be looking at the third digit of a three digit pair, and it must be the $608$th pair.

The $600$th pair would be $699$. So, the $608$th pair would be $707$. The third digit of this is a $7$, which would be the solution.