If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: 
$\textbf{(A)}\ \text{only }1\qquad \textbf{(B)}\ 1\text{ and }8\qquad \textbf{(C)}\ \text{only }8\qquad \textbf{(D)}\ 8\text{ and }49\qquad \textbf{(E)}\ 1,8,\text{ and }49$
We can plug in 1, 8, and 49 to see which works.
$1 = 1 = 1^2$
$1 + 2 + 3 + \cdots + 8 = \frac{8 \cdot 9}{2} = 36 = 6^2$
$1 + 2 + 3 + \cdots + 49 = \frac{49 \cdot 50}{2} = 1225 = 45^2$
All of these values produce a perfect square for $1 + 2 + \cdots + K,$ so the answer is $\boxed{\textbf{(E)}}.$
-coolmath34