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Motivated by this post and specially suggested by @Clement Yung:

Let $A$ and $B$ are strictly positive integers such that $A \geq B$.

Question: How to prove or make a counterexample for the following statement.

$$ \boxed{A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq \lfloor A/B \rfloor \times (A+1)} $$ Thanks in advance.

user0410
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1 Answers1

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Let $$\left\lfloor\frac AB\right\rfloor=k\ge1.$$

Then

$$A-k-\left\lceil\frac AB\right\rceil\le A-2k\le k(A+1).$$ is certainly true.