First, to the issue of "toxic" fields. To everyone out there who wrote that the idea is nonsense, there are people who are still working on elementary-ish approaches to special cases of Fermat's Last Theorem. There could be good math there, but if all I knew about somebody was that he submitted an elementary proof that $x^{17}+y^{17}\not=z^{17}$, I would be disimpressed, and suspect crank-hood.

There are also several comments about not putting it on your CV. If you make it onto a shortlist, you will be searched for at least through Google, the arXiv, and Math-SciNet. If it's out there, it will be attached to you. But also, if you have a couple of serious papers, and also one coauthored on a light topic, that would speak well of you (as a job candidate).

Have you considered the possibility that this professor knows his field better than you? Perhaps the result is interesting, but the question hadn't arisen before. There are, after all, problems that are easy to solve once they are stated in the right way. It's also possible that it's one of those problems that isn't too hard if it is in a book, and you know which chapter/section is relevant, but perhaps people in his field don't usually have your exact background. Perhaps this was the warm-up question, and for "the paper" he has some nice generalization or application in mind that still needs to be worked out.

The bottom line is that in math your name is your brand, and you need to nurture and develop and protect that brand. I know mathematicians that have pseudonyms that they use for exactly this class of publication: good enough to publish, not good enough to lump in with my "real" work.