2

For $A \geq B$, both are strictly positive integers, is the following true? $$A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq \lfloor A/B \rfloor \times (B+1)$$

I tried the technique used in proving a very similar question: Prove/Disprove: $A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq (\lfloor A/B \rfloor + 1) \times B$ for $A \geq B$

But it seems like it didn't work in proving this. I also tried empirically generating random A and B's, but also can't find a counterexample.

Nick
  • 167

1 Answers1

3

$\newcommand{f}[1]{\left\lfloor #1 \right\rfloor}$ Let $B = 100$ and $A = 199$. Then: \begin{align*} LHS &= 199 - 1 - 2 = 196 \\ RHS &= 1(100 + 1) = 101 \end{align*} So the inequality is false.

EDIT: In response to OP's comment, suppose we restrict further that $\f{A/B} \geq N$ for some $N \in \Bbb{Z}^+$. Let $B = 3N + 3$, and let $A = (N + 1)(3N + 3) - 1$. Clearly $A \geq B$ and $\f{A/B} = N$. \begin{align*} LHS &= (N + 1)(3N + 3) - 1 - N - (N + 1) \\ &= (N + 1)(3N + 1) \\ \end{align*} \begin{align*} RHS &= N(3N + 4) \\ &= N(3N + 1) + 3N \\ &= (N + 1)(3N + 1) - (3N + 1) + 3N \\ &= (N + 1)(3N + 1) - 1 < LHS \end{align*} So the inequality will still fail.

Clement Yung
  • 8,347
  • Oh! Thanks a lot! – Nick Aug 20 '20 at 10:57
  • Just curious, will it hold if $\lfloor A/B \rfloor \geq 2$? – Nick Aug 20 '20 at 11:05
  • What about the correctness of this version of the question? $$A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq \lfloor A/B \rfloor \times (A+1)$$ Thanks in advance – user0410 Aug 20 '20 at 11:07
  • @Nicholas see my edited answer. – Clement Yung Aug 20 '20 at 11:25
  • @user0410 I believe it's more appropriate to post that as a new question instead of commenting here. – Clement Yung Aug 20 '20 at 11:25
  • You know, I think it is not appropriate to post similar questions in SE especially when the base idea has been introduced by another OP. Therefore, maybe it is a suitable suggestion that please post another answer in this page and then kindly add more details about the comment. I appreciate you taking the time to answer our questions. – user0410 Aug 20 '20 at 11:42
  • @user0410 I get your point, but two questions being of a similar idea does not mean that they should be discussed in the same thread on stackexchange (since, in your case, they are still technically different questions). I suggest you credit the OP in your new question by linking to this thread, and I'll be happy to post an answer there after I think about it. – Clement Yung Aug 20 '20 at 11:45
  • Although I have bad experiences about these suggestions in SE (changed my username after five years and created user 0410), ok I will post a new questions. Thanks again. – user0410 Aug 20 '20 at 11:55
  • 1
    @user0410 I appreciate your understanding. – Clement Yung Aug 20 '20 at 11:56
  • Hi Clement, I've a follow-up question on this same question here: https://math.stackexchange.com/questions/3797474/how-does-a-relate-to-b-if-a-lfloor-a-b-rfloor-lceil-a-b-rceil-leq was wondering if you have any insights on it. – Nick Aug 20 '20 at 13:16