The integer $k$ is a [i]good number[/i], if we can divide a square into $k$ squares. How many good numbers not greater than $2006$ are there?

$ 
\textbf{(A)}\ 1003
\qquad\textbf{(B)}\ 1026
\qquad\textbf{(C)}\ 2000
\qquad\textbf{(D)}\ 2003
\qquad\textbf{(E)}\ 2004
$
1. **Understanding the problem**: We need to determine how many integers \( k \) (not greater than 2006) can be the number of squares into which a larger square can be divided. These integers are termed as "good numbers."

2. **Initial observations**:
   - We cannot divide a square into exactly 2 smaller squares.
   - We cannot divide a square into exactly 3 smaller squares.
   - We cannot divide a square into exactly 5 smaller squares.

3. **General case for odd \( k \) greater than 5**:
   - Let \( k = 2\ell + 5 \) where \( \ell \geq 1 \).
   - Divide the side length of the large square into \( \ell + 1 \) equal parts.
   - Fit \( \ell + 1 \) squares along one side and \( \ell \) more along a perpendicular side.
   - The remaining part is another square, which can be divided into 4 parts.
   - This gives \( 4 + \ell + 1 + \ell = 2\ell + 5 \) squares.

4. **General case for even \( k \) greater than 2**:
   - Let \( k = 2\ell + 2 \) where \( \ell \geq 1 \).
   - Divide the side length of the large square into \( \ell + 1 \) equal parts.
   - Fit \( \ell + 1 \) squares along one side and \( \ell \) more along a perpendicular side.
   - Leave the remaining square alone.
   - This gives \( 1 + \ell + 1 + \ell = 2\ell + 2 \) squares.

5. **Verification of small cases**:
   - \( k = 1 \): A single square is trivially a square.
   - \( k = 2 \): Impossible to divide a square into exactly 2 smaller squares.
   - \( k = 3 \): Impossible to divide a square into exactly 3 smaller squares.
   - \( k = 4 \): A square can be divided into 4 smaller squares (2x2 grid).
   - \( k = 5 \): Impossible to divide a square into exactly 5 smaller squares.

6. **Conclusion**:
   - The good numbers are all integers from 1 to 2006 except 2, 3, and 5.
   - Therefore, the number of good numbers is \( 2006 - 3 = 2003 \).

The final answer is \(\boxed{2003}\).