Will you get it's value?

Since $ax+b$ is the remainder of $f(x)$ when divided by $(x-91)(x-19)$ , we can write

$\begin{aligned} f(x) & = (x-91)(x-19) + ax + b \\ f(91) & = 91a + b = 19 & ...(1) \\ f(19) & = 19a + b = 91 & ...(2) \\ (1)-(2): \quad (91-19) a & = 19-91 \\ \implies a & = - 1 \\ (2): \quad -91 + b & = 19 \\ \implies b & = \boxed {110} \end{aligned}$

Generalization: We can generalize this by replacing $91$ and $19$ by $p$ and $q$ . Then we have:

If the remainder of $f(x)$ when divided by $(x-p)(x-q)$ is $ax+b$ , then $a=-1$ and $b=p+q$ .

$f(91)=19, f(19)=91\implies 19a+b=91, 91a+b=19\implies a=-1, b=\boxed {110}$ .