# Problem of the Week 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 $$\dfrac{8(9)}{2} = 36$$.