Say Laila gets a value of $x$ on her first 4 tests, and a value of $y$ on her last test. Thus, $4x+y=410.$
The value $y$ has to be greater than $82$, because otherwise she would receive the same score on her last test. Additionally, the greatest value for $y$ is $98$ (as $y=100$ would make $x$ as a decimal), so therefore, the greatest value $x$ can be is $98$. As a result, only $4$ numbers work, $86, 90, 94$ and $98$. Thus, the answer is $\boxed{\textbf{(A) }4}$.

1. Let $x$ be Laila's score on each of the first four tests, and let $y$ be her score on the last test. Given that her average score on the five tests was 82, we can set up the following equation:
$$\frac{4x + y}{5} = 82$$
2. Multiply both sides of the equation by 5 to clear the fraction:
$$4x + y = 410$$
3. We need to find integer solutions for $x$ and $y$ such that $0 \leq x \leq 100$ and $0 \leq y \leq 100$, with the additional condition that $y > x$.

4. Rearrange the equation to solve for $y$:
$$y = 410 - 4x$$
5. Since $y$ must be greater than $x$, we have:
$$410 - 4x > x$$
6. Combine like terms:
$$410 > 5x$$
7. Divide both sides by 5:
$$x < 82$$
8. Since $x$ must be an integer, the possible values for $x$ are $0, 1, 2, \ldots, 81$.

9. For each value of $x$, we calculate $y$ using $y = 410 - 4x$. We need $y$ to be an integer between 0 and 100, inclusive.

10. To find the range of $y$, we substitute the minimum and maximum values of $x$ into the equation $y = 410 - 4x$:
- When $x = 0$:
$$y = 410 - 4(0) = 410$$
This value is not within the range 0 to 100.
- When $x = 81$:
$$y = 410 - 4(81) = 410 - 324 = 86$$
This value is within the range 0 to 100.

11. We need to find the values of $y$ that are greater than $x$ and within the range 0 to 100. Since $y = 410 - 4x$, we need:
$$86 \leq y \leq 100$$
12. To find the corresponding values of $x$ for $y$ in this range, we solve:
- For $y = 86$:
$$86 = 410 - 4x \implies 4x = 324 \implies x = 81$$
- For $y = 90$:
$$90 = 410 - 4x \implies 4x = 320 \implies x = 80$$
- For $y = 94$:
$$94 = 410 - 4x \implies 4x = 316 \implies x = 79$$
- For $y = 98$:
$$98 = 410 - 4x \implies 4x = 312 \implies x = 78$$

13. Therefore, the possible values for $y$ are $86, 90, 94, 98$.

The final answer is $\boxed{4}$.