Answer
$$n \geq 16$$
Work Step by Step
Consider the error bound for Simpson's Rule:
$|E_S| \leq \dfrac{M(b-a)^5}{180n^4}$
The maximum value of $|f^{4}(x)|$ on $[1,3]$: $ M=24$
Now, $|E_S| \leq \dfrac{24 \times(3-1)^5}{180n^4}=\dfrac{32}{15n^4}$
We need to choose $n$ such that $\dfrac{64}{15n^4} \leq 10^{-4}$
This implies that $n \geq 14.4 \approx 15$
As per Simpson's Rule, we consider only the even value of $n$, so $n \geq 16$.
