This question comes from Grubb's Distributions and Operators, question 3.8(b):
Consider $u\in \mathcal{D}'(\mathbb{R}^n)$ and $\phi\in C^{\infty}_0(\mathbb{R}^n)$. Find out whether one of the following implications holds for arbitrary $u$ and $\phi$:$$ \langle u,\phi\rangle=0\Rightarrow \phi u=0 \:\:\:(1)$$or $$\phi u=0 \Rightarrow \langle u,\phi\rangle=0\:\:\: (2)$$
Not really sure how to approach this. My guess is that $(1)$ is more likely to hold than $(2)$ since there's at least an obvious example of the delta distribution, but I'm not sure how to go about proving/disproving either statement.