Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

$ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$
1. We are given that pigs are worth $ \$ 300 $ and goats are worth $ \$ 210 $. We need to find the smallest positive debt that can be resolved using these values.

2. To solve this, we need to find the smallest positive integer value that can be expressed as a linear combination of $ 300 $ and $ 210 $. This is equivalent to finding the smallest positive integer solution to the equation:
   \[
   300a + 210b = d
   \]
   where \( a \) and \( b \) are integers, and \( d \) is the debt.

3. The smallest positive value of \( d \) that can be expressed in this form is the greatest common divisor (GCD) of $ 300 $ and $ 210 $. We can find the GCD using the Euclidean algorithm:
   \[
   \begin{aligned}
   300 &= 210 \cdot 1 + 90, \\
   210 &= 90 \cdot 2 + 30, \\
   90 &= 30 \cdot 3 + 0.
   \end{aligned}
   \]
   The remainder becomes zero, and the last non-zero remainder is $ 30 $. Therefore, the GCD of $ 300 $ and $ 210 $ is $ 30 $.

4. This means that the smallest positive debt that can be resolved using pigs and goats is $ 30 $.

The final answer is \( \boxed{30} \).