The title pretty much explains this question. I am wondering what the name for the lemma that states "If one integer divides two different integers then it will divide their difference." Is it alright for me to ask this on Math Stack Exchange, or is there a better place to ask it? Thank you in advance!
Yes, these questions are alright. You can use the tag reference-request. The description of this tag is-
This tag is for questions seeking external references (books, articles, etc.) about a particular subject.
So if you want to ask a question regarding the name of a lemma, just post it by starting the question with some background: why do you need the lemma name, what have you researched, and where are you stuck? Then the question would most likely be fine.
I think such a question needs to justify its existence by stating why it is expected a name exists. If it does this, then it should be OK.
The issue is that most theorems (and almost all lemmas) do not have names - only the very famous ones do*. This means that these questions can be unanswerable: noone knows if a name exists, but noone is confident enough to say "there is no name".
*Naming theorems can result in confusion: As Jean-Claude Arbaut pointed out in the comments to the other answer, some theorems have multiple names, such as the squeeze theorem, while some names refer to multiple theorems, like "Cauchy's Theorem". Some results are even miss-attributed, for example people still refer to The Lemma That Is Not Burnside's as "Burnside's Lemma", including Wikipedia.