Let $S = \{n ∈ N | n \text{ divides the sum of any n consecutive numbers} \}$.
How can I describe the set S? I was given the hint: $\displaystyle\sum\limits_{n=1}^N n=\frac{N(N+1)}{2}$
I don't want this solved for me, but I don't understand how the hint works for this, because what if $N=3$, then the formula for $n$ doesn't hold for all $n$ between $1$ and $N$, not even $n=1$.
Thanks for any help!
Hint: Consider any run of $n$ consecutive integers, say: $$ a + 1, a + 2, \ldots, a + n $$ Summing them together, we have: $$ \sum_{k=1}^n (a + k) = \sum_{k=1}^n a + \sum_{k=1}^n k = an + \frac{n(n + 1)}{2} = n\left( a + \frac{n + 1}{2}\right) $$ Now for what values of $n$ will the expression in brackets be an integer?