For some integers a and b, the same integer satisfies both of the equations, what is the greatest possible value? For some integers $a$ and $b$, the same integer satisfies both of the equations $$x^2 - ax + 2014 = 0$$ and $$x^2 - bx + 2015 = 0.$$ What is the greatest possible value of this common solution?
Let $x_1$ and $x_2$ be roots of the first equation and $x_1$ and $x_3$ be roots of the second.

Thus, $x_1x_2=2014$ and $x_1x_3=2015$, which gives $2014$ and $2015$ divided by $x_1$,

which gives $x_1=1$ or $x_1=-1$, which gives the answer: $1$.
Both solutions are integers (this is the big hint), so let factoring them out
For x^2 - ax + 2014,there are only two possible factors
(x±1)(x±2014) or (x±2)(x±1007)
the solutions will be ±1 &amp; ±2014, or ±2 &amp; ±1007

For x^2 - bx + 2015, 
(x±1)(x±2015) or (x±5)(x±403) the solutions are ±1 &amp; ±2015 or ±5 &amp; ±403

So, the greatest possible common solution is 1