Mr. Sanchez's students were asked to add two positive integers. Juan subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the correct answer?
Call the two positive integers $x$ and $y$. Without loss of generality, assume $x > y$. We can write a system of equations to represent the information given in the problem:  \begin{align*}
x - y &= 2 \\
x \cdot y &= 120
\end{align*} Solving for $x$ in the first equation yields $x = y + 2$.

Substituting this into the second equation gives $(y + 2) \cdot y = 120$, or $y^2 + 2y - 120 = 0$.

This quadratic equation factors into $(y + 12)(y-10) = 0$, so $y = 10$.

Given $y$, we can solve for $x$ to get $x = 12$, so $x + y = \boxed{22}$.