Problem E

All Square

The positive multiples of \(3\) from \(3\) to \(2400\), inclusive, are each multiplied by the same positive integer, \(n\). All of the products are then added together and the resulting sum is a perfect square.

Determine the value of the smallest positive integer \(n\) that makes this true.

Note: In solving this problem, it may be helpful to use the fact that the sum of the first \(n\) positive integers is equal to \(\dfrac{n(n+1)}{2}\).

That is, \[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\] For example, \(1 + 2 + 3 + 4 + 5 = 15\), and \(\dfrac{5(6)}{2} = 15.\)

Also, \(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36\) and \(\dfrac{8(9)}{2} = 36\).