The quadratic equation $x^2-mx+24 = 10$ has roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?

Simplifying gives x^2 - mx + 14 = 0. By Vieta's formula, x_1 x_2 = 14 and m = x_1 + x_2. Which integers multiply to 14? (-7, -2), (-14, -1), (7, 2), and (14, 1), assuming |x_1| > |x_2|. So the possible values of m are -9, 9, -15, and 15.