The following proposition (not as a proposition but merely as a one-line fact) was stated in a paper that I have been studying as ‘easily seen’ to hold, while I, unfortunately, could not see it so easily! Nonetheless, I think it is a nice result.
For a positive integer, we use the usual notation to denote the largest integer such that and . (Apparently, this is called the -adic order, I still haven’t figured out why.) Further, we write if .
Proposition. Let be a prime number and positive integers. Let be an integer with that is maximally divisible by , i.e., with . Then, if
We will use the following lemma which is easy to prove.
Lemma. For all real numbers ,
where is the fractional part of .
Proof of the proposition. Let . Then
First suppose that . Then , so by the lemma we get
as desired. Now suppose that for some . Then , so , for each . Thus
Hence the result follows.