Floor Function Bound?

Question

I am trying to prove or disprove the following bound:

$2+\left(n-\left\lfloor\dfrac n k\right\rfloor k\right)\ge \left\lfloor\dfrac n k\right\rfloor$, where $2<k\le \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k, n$ are coprime, and $n,k\in\mathbb N$.

Any suggestions, solutions, or hints would be appreciated!

Answer 1

It is not true. Take $n$ huge. The left side is no greater than $k+1$, the right is unbounded in $n$. For a specific, $n=101, k=3$, we are comparing $2+(101-33\cdot 3)=4$ and $\lfloor \frac {101}3 \rfloor =33$

Answer 2

Hint: $n=16$, $k=3$ and $n=7$, $k=3$. The first case yields $<$, the second one yields $>$.