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.
$\newcommand{f}[1]{\left\lfloor #1 \right\rfloor}$ 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.
