$ \text{Let } S_K=\dfrac{1+2+\cdots+K}{K} \text{ and } \displaystyle\sum_{j=1}^{n} S_j^{2}=\dfrac{n}{A}\big(Bn^{2}+Cn+D\big),\ \text{where } A,B,C,D\in\mathbb{N} \text{ and } A \text{ has least value. Then:} $