Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.
1. We start with the given equation:
   \[
   2a - 9b + 18ab = 2018
   \]

2. Move \(2018\) to the left-hand side:
   \[
   18ab + 2a - 9b - 2018 = 0
   \]

3. To factor this expression, we use Simon's Favorite Factoring Trick. We add and subtract 1 to the left-hand side:
   \[
   18ab + 2a - 9b - 2018 + 1 - 1 = 0
   \]
   \[
   18ab + 2a - 9b - 2017 - 1 = 0
   \]
   \[
   18ab + 2a - 9b - 2017 = 1
   \]

4. Now, we can factor the left-hand side:
   \[
   (9b + 1)(2a - 1) = 2017
   \]

5. We need to find the prime factorization of \(2017\). Since \(2017\) is a prime number, its only factors are \(1\) and \(2017\).

6. We consider the possible factor pairs of \(2017\):
   \[
   (9b + 1)(2a - 1) = 2017 \cdot 1
   \]

7. Solving for \(b\) and \(a\) from the factor pairs:
   - If \(9b + 1 = 2017\) and \(2a - 1 = 1\):
     \[
     9b + 1 = 2017 \implies 9b = 2016 \implies b = \frac{2016}{9} = 224
     \]
     \[
     2a - 1 = 1 \implies 2a = 2 \implies a = 1
     \]

8. Therefore, we have \(a = 1\) and \(b = 224\).

9. Finally, we calculate \(b - a\):
   \[
   b - a = 224 - 1 = 223
   \]

The final answer is \(\boxed{223}\).