Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18$
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 \(\boxed{4}\).