Factorization by inspection

Observe that polynomial equation of degree $16$ $f(x) = x+1$ has exactly $16$ roots: $x = 4, 5, 6, \dots, 18, 19.$ And by the fact that $f(x)$ is monic (i.e., leading coefficient is $1$ ), we have the factorization $f(x) - (x+1) = (x-4)(x-5)(x-6) \cdots (x-18)(x-19),$ namely, $f(x) = (x+1) + (x-4)(x-5)(x-6) \cdots (x-18)(x-19).$

Next, notice that $f(20) = 21 + 16 \cdot 15 \cdot 14 \cdots 2 \cdot 1 = 21 + 16!.$

Now that $17$ is a prime, and by Wilson’s theorem , we then have $16! \equiv -1 \pmod{17}.$ Thus, $\begin{aligned} f(20) &= 21 + 16! \\ &\equiv 21 + (-1) \pmod{17} \\ &= 20 \\ &= \boxed{\mathbf{3}} \pmod{17}. \end{aligned}$