Let $p$ be an integer such that both roots of the equation $5x^2 - 5px + (66p-1) = 0$ are positive integers. Then the value of $\left\lfloor\frac{p}{10}\right\rfloor$ is equal to ($\lfloor . \rfloor$ denotes greatest integer function)
Step-by-Step Solution
Key Concept: Difference of squares factorization converts a Diophantine equation into a system of factor pair constraints.
From $p^2 - 20(6p-1) = k^2$, we get $(5p-132)^2 - k^2 = 17404$, so $(5p-132-k)(5p-132+k) = 2^2 \times 10 \times 229$. Testing factor pairs and requiring $p$ to be an integer, we find $5p - 132 - k = 2 \times 19$ and $5p - 132 + k = 2 \times 229$, yielding $p = 76$.
Correct Answer: 7