2002 AMC 10A Problems/Problem 14
Problem
The 2 roots of the quadratic 
are both prime. How many values of k are there?
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E)}&$ (Error compiling LaTeX. Unknown error_msg)More than 4
Solution
Consider a general quadratic with the coefficient of 
as one and the roots as r and s. It can be factored as 
which is just 
. Thus, the sum of the roots is the negative of the coefficient of x and the product is the constant term. (In general, this leads to Vieta's Formulas).
We now have that the sum of the two roots is 63 while the product is k. Since both roots are primes, one must be 2, otherwise the sum is even. That means the other root is 61 and the product must be 122. Hence, our answer is 
.
