The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those $C^{\infty }$ multipliers F such that the division problem is solvable in $S^{\prime }(\mathbb{R})$. For these functions $F\in O_M(\mathbb{R})$ we even prove that the multiplication operator $M_F(G)=FG$ has a continuous linear right inverse on $S^{\prime }(\mathbb{R})$, in contrast to what happens in the several variables case, as was shown by Langenbruch.