Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$
I notice that the equation has integer coefficients, so if the roots are prime numbers, they must be factors of the constant term $k.$ I also notice that the sum of the roots is $63,$ which is an odd number. This means that one root must be odd and the other even, and the only even prime number is $2.$ Therefore, one root is $2$ and the other is $61,$ which is also prime. This implies that $k = 2 \times 61 = 122.$ There is no other way to write $k$ as a product of two prime numbers that add up to $63,$ so this is the only possible value of $k.$ 