Since the average score on the five tests is $82,$ the total score of those five tests must be $5\cdot82 = 410 .$

Now, let $f$ be the score on the first 4 tests and let $l$ be the score for the last test.

We know that $$f < l \leq 100$$ and $$4f + l = 410.$$ And as $410 = 4f + l < 5l ,$ we know $\frac{410}{5} = 82 < l .$

Also, since $4f + l = 410 ,$ and dividing $410$ by $4$ gives us a remainder of 2, we know that dividing $l$ by $4$ must leave a remainder of $2$ as $4f$ will leave no remainder when divided by $4.$ Equivalently: $$l \equiv 2 \mod 4 .$$ Since $82 < l \leq 100$ and $l \equiv 2 \mod 4,$ the only options for $l$ are $86,90,94,98.$ This yields four distinct solutions as follows: $$\begin{align*} (f,l) =& (81,86);\\ &(80,90);\\ &(79,94);\\ &(78,98) \end{align*}$$ Therefore, there are $4$ solutions, and A is the correct answer.