A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ however its digits are rearranged. How many good numbers are there?

$ 
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{Infinitely many}
$
To determine how many good numbers there are, we need to understand the properties of such numbers. A good number \( n \) is defined such that every integer multiple of \( n \) remains divisible by \( n \) even if its digits are rearranged.

1. **Understanding the properties of good numbers:**
   - A number \( n \) is good if for any multiple \( mn \) of \( n \), any rearrangement of the digits of \( mn \) is also divisible by \( n \).
   - This implies that \( n \) must be such that the sum of the digits of any multiple of \( n \) is divisible by \( n \). This is because the sum of the digits of a number is invariant under digit rearrangement.

2. **Considering the sum of digits:**
   - For a number \( n \) to be good, the sum of the digits of any multiple of \( n \) must be divisible by \( n \).
   - This property is satisfied if \( n \) divides 9. This is because the sum of the digits of a number modulo 9 is congruent to the number itself modulo 9 (a property of numbers in base 10).

3. **Finding the divisors of 9:**
   - The divisors of 9 are 1, 3, and 9.
   - We need to check if these numbers are indeed good numbers.

4. **Verification:**
   - For \( n = 1 \): Any number is divisible by 1, and any rearrangement of its digits is also divisible by 1. Hence, 1 is a good number.
   - For \( n = 3 \): A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Any rearrangement of the digits of a multiple of 3 will still have a sum of digits divisible by 3. Hence, 3 is a good number.
   - For \( n = 9 \): A number is divisible by 9 if and only if the sum of its digits is divisible by 9. Any rearrangement of the digits of a multiple of 9 will still have a sum of digits divisible by 9. Hence, 9 is a good number.

Since 1, 3, and 9 are the only divisors of 9, and they all satisfy the condition of being good numbers, we conclude that there are exactly 3 good numbers.

The final answer is \(\boxed{3}\)