# Prove/Disprove: $A - \lfloor A/B \rfloor - \lceil A/B \rceil \leq \lfloor A/B \rfloor \times (B+1)$ for $A \geq B$

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.

• Oh! Thanks a lot!
– Nick
Commented Aug 20, 2020 at 10:57
• Just curious, will it hold if $\lfloor A/B \rfloor \geq 2$?
– Nick
Commented Aug 20, 2020 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 Commented Aug 20, 2020 at 11:07
• @Nicholas see my edited answer. Commented Aug 20, 2020 at 11:25
• @user0410 I appreciate your understanding. Commented Aug 20, 2020 at 11:56