one of m actors is 4

Let's break down the clues to find the mystery number.

1. One of the factors is 4.
2. When paired with 32, the LCM (Least Common Multiple) is 32.
3. The mystery number is greater than 4 and less than 15.

Using the second clue regarding the LCM:

- The LCM of two numbers \( a \) and \( b \) can be calculated using the formula:
\[
\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}
\]

Since the LCM of the mystery number \( x \) and 32 is 32, it means \( x \) must be a factor of 32 that is also greater than 4 and less than 15.

The factors of 32 are: 1, 2, 4, 8, 16, and 32.

From these, the factors that are greater than 4 and less than 15 are:
- 8

Now let’s check the LCM of 8 and 32:
- The GCD of 8 and 32 is 8.
- Therefore, the LCM of 8 and 32 is:
\[
\text{LCM}(8, 32) = \frac{8 \times 32}{8} = 32
\]

This confirms that the conditions are satisfied.

Thus, the mystery number is **8**.