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

,

then .

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.

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