Submitting for publication in American Mathematical Monthly I am a young computer science researcher. One of my working papers had some mathematical content so I presented it in a math department seminar. After the talk, one of the professors told me that he liked the talk and suggested that I submit the paper to American Mathematical Monthly. I was flattered, since this journal published many of the important results in my field (as well as in other fields, of course).
Then, another professor who overheard the conversation said to his friend "do not do this to him! He is still young!". I was perplexed and did not have time to ask: why is the American Mathematical Monthly not a good option for a young researcher?


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*Is it because it is too competitive, and I may wait a long time only to get rejected? But how is it different than other competitive journals? Does it have special requirements that are particularly hard to get by young researchers?

*Is it because it's impact factor is low, so it is not sufficiently valued by faculty promotion committees?

 A: Arguably this question would be better posted on academia.SE, but by posting it here I suppose you want to ensure a math-specific answer.  (Okay!)
The beginning and end of the answer has got to be for you to talk to one or both of these professors and also your thesis advisor to see what they have in mind.  
Anyway: the American Mathematical Monthly (henceforth the Monthly) is really one of a kind when it comes to mathematics journals.  Here are some points of description:


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*It is not a research journal — it is published by the Mathematics Association of America, whose goals concern teaching and exposition.  On the other hand, the majority of papers published there have some research content, and a good percentage of the papers published there (maybe 10 per year) have substantial research content, by which I mean that they are of more interest to the mathematics research community than the lowest tiers of research journal.

*It is very competitive to get published in the Monthly: I believe their acceptance rate is approximately 10–20% (I haven't checked into this recently, but I know several editors and could do so), which is much more competitive than the lowest tiers of research journals.  A substantial part of the "competition" comes from the quality of the exposition: they have extremely high expository standards.  See this closely related answer of mine for more verbiage on this point.

*With respect to the average in (pure) mathematics, they are rather fast in processing papers: the current editor is admirably meticulous and efficient in processing of submissions, and referees are typically given no more than three months to write a report.  They do solicit reports from two referees, which raises the chances of something slowing down the process.

*Of course they do not have more stringent requirements from young researchers.  I know of no journal that does.  In fact for several years the Monthly has had a double blind submission process, meaning that the author and the referees are anonymous to each other.  This is all but unheard of in pure mathematics.  (The only other double blind pure math journals that I know of are the other MAA journals...but at present they are not publishing even halfway serious papers, and some of them have other problems.  I do not recommend submitting there.)  I believe it can sometimes be slightly easier for younger researchers to get published in a single blind system: professionals can get held to more stringent standards as the years roll by...but presumably this happens less in a double blind system.

*The Monthly has, by far, the highest readership of any reputable mathematics journal.  More than that, it is one of very few mathematics journals that actually has a readership in the usual sense: i.e., there are many mathematicians who subscribe to it and peruse every issue.  

*The preceding bullet point is possible because the Monthly targets papers that will be accessible to the widest possible mathematical audience.  Graduate level mathematics can appear in Monthly articles, but it must be very carefully introduced, motivated and explained.  

*Because of the above points, the Monthly is quite a prestigious journal...but in its own unique way.  As you say, its impact factor is low.  (But the impact factor is still not taken so seriously in pure mathematics.  Anyway, how to measure the impact factor of the Monthly depends upon what you count as articles.  If you look on MathSciNet, you'll see that the recorded number of articles published per year has wild variation.  If you were to measure the impact factor with respect to full length articles only — which comprise about half the contents of any given issue — I think it would be more respectable.)  In the United States, for a mathematician who is employed at a more "teaching focused institution," a Monthly publication would probably confer substantial cachet.  For a more research focused mathematician, a Monthly publication still looks nice but is of less clear benefit.  
Let me speak personally for a little while: I have submitted five pieces to the Monthly, of which three have been published.  All three publications have multiple citations.  However, none of them concern my main research interests, and two of them have nothing to do with any other papers I have written.  I am happy to have these publications and look forward to more in the future, but the amount of time I spent on each of them — per page or per mathematical research unit — was more than on most of my other papers.  Each time I had to make substantial revisions of a sort that a research journal would probably not have required.  
Now back to you: you should be aware that the effort required to publish your paper in the Monthly may be disproportionate to what you gain from it professionally.  I would also think very carefully about publishing a result from the mainstream of your research in the Monthly: to the best of my knowledge, that is very rarely done nowadays.  I can see (at least) the following risks for doing so:
(i) If you have an important contribution to your subfield of research, then it is likely that you will want to write it up for a target audience of other researchers in your subfield.  That is really antithetical to the way a Monthly publication is written up.  If you already have some rough writeup for the former audience, it will take you a lot of time and work to get a Monthly appropriate writeup.  You should also ask yourself whether a broadly pitched piece will be received more favorably by those in your subfield.
(ii) I can think of very few (less than five) Monthly publications in recent years that feature true research breakthroughs (i.e., the kind of paper that deserves to be published in a top tier research journal: note well that in between the top tier and the lowest tiers lie a lot of other tiers).  By publishing a paper in the Monthly you are probably not signalling to those who evaluate your work that you want your paper to be viewed in this way.  I do notice that you work in the field of fair division in which several important papers have been published in the Monthly.  For instance, there is a 1999 paper of F. Su, with 33 MathSciNet citations.  This is an interesting phenomenon, and as an outsider to your subfield it's really not up to me to evaluate it.  Maybe those who work in fair division view the Monthly as a place where top research papers are published. If I had to guess — I doubt it.  But don't make me guess — ask your advisor and the other professors you spoke to!  
To sum up a long answer: I can't properly judge, but I suspect that you don't want to publish your latest nice piece of research in the Monthly.  If that turns out to be the case, I suggest that you keep the Monthly in mind: perhaps in a few years you will want to publish a broader, more expository paper that intersects your current paper in content.  (By the way, Monthly papers do not need to have new results, although the good ones offer at least novel perspectives.)  Good luck.
