Remainder in polynomial division The remainder when $x^{50}$ is divided by $(x-3)(x+2)$ is of the form $ax + b$. Find the units digit of $a$.
I tried to tackle the problem using the polynomial remainder theorem but got stuck as the divisor is a quadratic expression. 

Hint
$x^{50}=(x-3)(x+2)q(x)+ax+b$ (Division algorithm)
Now, take $x=3$ and $x=-2$ to get two equations. Two variables and two equations. Hope you get it.
This is the complete problem. (Use the hint and try yourself first)
Taking $x=3$,
$3^{50}=3a+b,$
Taking $x=-2$
$(-2)^{50}=2^{50}=-2a+b$
Subtracting, we get
$3^{50}-2^{50}=5a\implies 9^{25}-4^{25}=5a$
Firstly, observe that the LHS is divisible by $(9-4)=5$(Why?). So, you get an integer value of $a$. (Just for a check)
$\therefore a= 9^{24}+9^{23}\cdot 4+ 9^{22}\cdot 4^2+...+4^{24}$
Now, you may use modular arithmetic. 
$a\equiv 1-4+6-4+6-4+...+6 \equiv 1+12\times 2\equiv 5\pmod{10}$ (Why?)
Hope you get it.
