I actually think it will not be so easy to say when two proofs of this result will be "distinctly different": rather I expect most or all will have common features, including using at least a little bit of group theory.
For instance, the proof I wrote up for my elementary(ish) number theory course is Theorem 9 in these notes. The notes themselves are on finite commutative groups, and Theorem 9 is on page 3, in the section on "cyclic groups". Prior to the statement and proof, a little over a page is spent developing the basic properties of cyclic groups, including a statement involving the Euler $\varphi$-function. The proof of the result itself -- which, note, is a criterion for an a priori noncommutative finite group to be cyclic -- occupies $11$ lines. (Added: sorry, false advertising -- add two more lines to get from Theorem 9 to Corollary 10, which is the statement that any finite subgroup of the multiplicative group of a field is cyclic.) I certainly think it is more or less the proof that any research mathematician is expecting to find.
Let me mention though that I had originally included this argument as an application of the Mobius Inversion Formula. After having looked back at what I'd done, I decided that although the argument was reminiscent of an inversion / inclusion-exclusion counting argument, it only made it more complicated to phrase it in that way.