Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
1. **Volume Calculation**:
   To find the smallest positive integer \( n \) such that a cube with sides of length \( n \) can be divided into exactly 2007 smaller cubes, each of whose sides is of integer length, we start by considering the volume relationship. The volume of the larger cube is \( n^3 \), and the total volume of the 2007 smaller cubes must also be \( n^3 \).

   \[
   n^3 \geq 2007
   \]

2. **Finding the Minimum \( n \)**:
   We need to find the smallest integer \( n \) such that \( n^3 \geq 2007 \). Calculating the cube root of 2007:

   \[
   \sqrt[3]{2007} \approx 12.6
   \]

   Since \( n \) must be an integer, we take the next whole number:

   \[
   n \geq 13
   \]

3. **Verification for \( n = 13 \)**:
   We need to verify if \( n = 13 \) can be divided into exactly 2007 smaller cubes. First, calculate the volume of the cube with side length 13:

   \[
   13^3 = 2197
   \]

   We need to check if we can partition this volume into 2007 smaller cubes with integer side lengths. One possible way to do this is to use a combination of cubes of different sizes.

4. **Partitioning the Volume**:
   Let's consider the partition:

   \[
   13^3 = 3 \cdot 3^3 + 16 \cdot 2^3 + 1988 \cdot 1^3
   \]

   - \( 3 \cdot 3^3 \) represents three 3x3x3 cubes.
   - \( 16 \cdot 2^3 \) represents sixteen 2x2x2 cubes.
   - \( 1988 \cdot 1^3 \) represents 1988 1x1x1 cubes.

   Calculate the total number of smaller cubes:

   \[
   3 + 16 + 1988 = 2007
   \]

   This confirms that the partition is correct.

5. **Other Possible Partitions**:
   The solution also mentions other ways to dissect the cube of length 13, such as:
   - {1x5, 2x3, 2x2, 2002x1}
   - {1x4, 3x3, 7x2, 1996x1}

   These partitions also satisfy the condition of having 2007 smaller cubes.

The final answer is \( \boxed{ n = 13 } \).