I encountered this problem while solving a problem related to characterization of delta function upto constant multiple.
$\phi \in D(R)$, Space of compactly supported infinitely differentiable functions. If $\phi(0)=0$, Then there exist a $\psi \in D(R)$ such that $\phi = x\psi$
Original question is as follow:
If $xT=0$, where T is a distribution. Then, $T = c\delta$
If above proposition could be proved, then this characterization would follow.