### Instruction: Let $f(x) = 12x+7$ and $g(x) = 5x+2$ whenever $x$ is a positive integer. Define $h(x)$ to be the greatest common divisor of $f(x)$ and $g(x)$. What is the sum of all possible values of $h(x)$?### Response: I want to find all possible values of $h(x)$, which is the greatest common divisor of $f(x)$ and $g(x)$.
I know that the greatest common divisor of two numbers is the largest positive integer that divides both numbers evenly.
So, I need to look at the factors of $f(x)$ and $g(x)$ and find the largest one they have in common.
To do that, it might help to write $f(x)$ and $g(x)$ in factored form, using prime factorization.
I can use the distributive property to write $f(x) = 12x+7 = 3 \cdot 4 \cdot x + 7$ and $g(x) = 5x+2 = 5 \cdot x + 2$.
Now, I can see that $f(x)$ and $g(x)$ have no common factors when $x$ is a positive integer, except for $1$.
That means that $h(x) = 1$ for any positive integer $x$.
So, the only possible value of $h(x)$ is $1$, and the sum of all possible values of $h(x)$ is just $1$.