Find an integer, n, greater than 4 where the sum of n consecutive integers is divisibly by n?

Question

Find an integer, n, greater than 4 where the sum of n consecutive integers is divisibly by n

Answer 1

The "n" consecutive integers starting at x are

x + 0, x+1, x+2, x+3, ..., x+n-1

That equals n*x + sum (1..n-1)

sum (1..n-1) = (n-1)(n)/2

That is divisible by n if (n-1) is even, which means n is odd.

It works for all odd n.

Another way to look at it is:

take x to be the middle number.

Below x you have x - 1, x - 2, ... x-k/2

Above x you have x + 1, x + 2, ... x+k/2

If you add all that up, you get 2k*x

add in the middle x and you (2k+1)x

Let n = 2k+1, which is odd, and you have it.

It doesn't work if n is even, because there will be

one more to one side or the other and the sum

will n * x + (something < n).

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