How will you convert methyl bromide to dimethylamine?
Well, you can do this in one step; a methyl bromide is not very sterically hindered at all, so a single
For a solvent you can use an aprotic solvent like DMSO (dimethylsulfoxide), or perhaps acetone, which does not "distract" the methylamine by inviting it to come get a proton. (That would have deactivated it and then we wouldn't have a nucleophile.)
By the time the reaction has finished, either the methylamine nucleophile or another molecule of the dimethylamine product would have equilibrated to acquire the proton from the protonated product, since the pKa of methylamine is about
The equilibrium lies on the side of the weaker acid.