For some integers a and b, the same integer satisfies both of the equations, what is the greatest possible value?

Question

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?

Answer 1

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: $1$.