Both roots of the quadratic equation $x^2 - 63 x + k = 0$ are prime numbers. How many possible values of $k$ are there?
I recall that the sum of the roots of a quadratic equation is equal to the opposite of the coefficient of $x$, and the product of the roots is equal to the constant term. So, if I let $p$ and $q$ be the prime roots of the equation, I have $p + q = 63$ and $pq = k$. The first equation implies that one of the roots must be even, since the sum of two odd numbers is even. The only even prime number is $2$, so one of the roots must be $2$. That means the other root must be $63 - 2 = 61$, which is also prime. So, the only possible pair of prime roots is $(2, 61)$. Therefore, the only possible value of $k$ is $2 \times 61 = 122$. 