please help

\(x^2-ax+2016=0\)

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 \(\alpha < \beta\).

 

Then, 

\(x^2 - ax + 2016 = (x- \alpha)(x - \beta)\\ x^2 - ax + 2016 = x^2 - (\alpha + \beta)x + \alpha \beta\)

 

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:

\(\alpha\)	1	2	3	4	6	7	8	9	12	14	16	18	21	24	28	32	36	42
\(\beta\)	2016	1008	672	504	336	288	252	224	168	144	126	112	96	84	72	63	56	48

 

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.