The quadratic equation x^2 - ax + 2016 = 0 has two postiive integer solutions. Find the minimum value of a.
fixed your LaTeX
(it was kind of weird to read)
We let the positive integer solutions to be \(\alpha\) and \(\beta\). Without loss of generality, assume \(\alpha < \beta\).
\(x^2 - ax + 2016 = (x- \alpha)(x - \beta)\\ x^2 - ax + 2016 = x^2 - (\alpha + \beta)x + \alpha \beta\)
So, the question is actually asking for the minimum value of \(\alpha + \beta\) for \(\alpha \beta = 2016\). We can list the factors of 2016.
Factors of 2016:
I have listed them in a unique way such that on each column, the numbers multiply to 2016.
We can see that a is the minimum sum of these pair of numbers, which is 42 + 48 = 90.