Let $f(x)=3x+2$ and $g(x)=ax+b$, for some constants $a$ and $b$. If $ab=20$ and $f(g(x))=g(f(x))$ for $x=0,1,2...9$ find the sum of all possible values of $a$.

For nested functions, we take the interior function and substitute it into $x$ of the outer function.

$f(g(x)) = 3ax + 3b + 2$

$g(f(x)) = 3ax + 2a + b$

$3ax + 3b + 2 = 3ax + 2a + b \rightarrow 2a = 2b+2$

Thus, we need to solve the system of equations

$ab = 20$

$2a = 2b + 2$

We can either solve for a variable and substitute into the other equation, or we can guess and check to see that our solutions are

$a = 5, b = 4$

$a = -4, b= -5$

Thus the sum of all possible values of $a$ is $5 - 4 = \boxed{1}$.