Answer: 987

Small Hint:

Modulo $x^2-x-1,$ powers satisfy $x^n=F_nx+F_{n-1}$

Big Hint:

Set both coefficients of the linear remainder equal to zero and use Cassini’s identity

Solution:

Modulo $x^2-x-1,$ the relation $x^2=x+1$ gives $x^n=F_nx+F_{n-1}.$ Hence the remainder is $$\begin{aligned}r(x)={}&(1597a+987b)x\\&+(987a+610b+1).\end{aligned}$$ Both coefficients must vanish. The determinant of the coefficient matrix is $$1597(610)-987^2=1$$ by Cassini’s identity. Cramer’s rule then gives $a=987.$