jmc

The first answer choice $\text{(A)}3$, can be eliminated since there must be $10$ squares with integer side lengths. We then test the next smallest sidelength which is $4$. The square with area $16$ can be partitioned into $8$ squares with area $1$ and two squares with area $4$, which satisfies all the conditions of the problem. Therefore, the smallest possible value of the length of the side of the original square is $\boxed{4}$.