jmc

Since we are dividing by a quadratic, our remainder will have degree at most $1$. Hence the remainder is of the form $ax+b$ for some constants $a$ and $b$. We have $$P(x)=(x-13)(x-17)Q(x)+ax+b$$where $Q(x)$ is the quotient when $P(x)$ is divided by $(x-13)(x-17)$. We can eliminate the $Q(x)$ term by plugging in $x=13$ or $x=17$. Using the Remainder Theorem, each gives us an equation: $$P(13)=13a+b=6$$$$P(17)=17a+b=14$$Solving this system gives us $a=2$ and $b=-20$, and hence the remainder when $P(x)$ is divided by $(x-13)(x-17)$ is $\boxed{2x-20}$.