The question i was given by a friend is stated as follows. I think there is an error.
Write the first n positive integers from left to right in their natural order to form a large integer: $$123456789101112 · · · n.$$ Suppose that this large number has 6927 digits. Find the value of n. Justify your answer.
So counting off the first 9 digits leaves $6918$ digits remaining.
Counting off the next $180=2\cdot 90$ digits leaves $6738$ digits remaining.
Counting off the next $2,700=3\cdot 900$ digits leaves $4038$ digits remaining.
There is my problem. So $n$ has to be a $4$-digit number, and so dividing by $4000$ tells me that i only have 38 digits to go, and so the number has to be 2009, but this would require 40 digits not 38.
I would thing then maybe they were asking for the actual last number (1's digit) in this huge integer, but the fact that we are writing the first $n$ number seequentially, and since $4038$ is not divisible by $4$ makes me think this problem is incorrect. Thoughts?
EDIT: So i guess my confusion is over the $n$. Is $n=0$ or are they asking for the $n$th digit string (2009) here?
Or are they asking for n such that "6927" is within the resulting number?
For that interpretation, I think the lowest n is 270 (n = 927 works also).