Wipeout

Take the numbers $1, 2, 3, 4, 5, 6$ and choose one to wipe out.

For example, you might wipe out $5$, leaving you with $1, 2, 3, 4, 6$.

The mean of what is left is $3.2$.

I wonder whether I can wipe out one number from $1$ to $6$, and leave behind an average which is a whole number... 

What about starting with other sets of numbers from $1$ to $N$, where $N$ is even, wiping out just one number, and finding the mean?

Which numbers can be wiped out, so that the mean of what is left is a whole number? Can you explain why?

What happens when $N$ is odd?

Here are some puzzling wipeouts you might like to try:

One of the numbers from $1$ to $15$ is wiped out.
The mean of what is left is $7.\dot{7}1428\dot{5}$
Which number was crossed out?

One of the numbers from $1$ to $N$, where $N$ is an unknown number, is wiped out.
The mean of what is left is $6.8\dot{3}$
What is $N$, and which number was crossed out?

One of the numbers from $1$ to $N$ is wiped out.
The mean of what is left is $25.76$
What is $N$, and which number was crossed out?