<PRE><B>Question 485869</B>
We can say that


*[tex \LARGE a + (a+1) + ... + (a+n-1) = 585] where a and n are integers. This simplifies to


*[tex \LARGE an + (1+2+...+(n-1)) = 585]


*[tex \LARGE an + \frac{(n-1)(n)}{2} = 585]


*[tex \LARGE n(a + \frac{n-1}{2}) = 585]


If n = 585, then (n-1)/2 = 292 and we can set a = -291. We can check that (-291) + (-290) + ... + 291 + 292 + 293 = 585, because all the terms except 292 and 293 cancel (and 292 + 293 is obviously 585). Hence n = 585 is the greatest such number.</PRE>