I'm aware that if $$$2x \leq y$$$ then $$$x \leq \lfloor y/2 \rfloor$$$, and that if $$$2x \geq y$$$ then $$$x \geq \lceil y/2 \rceil$$$

I usually remember that by writing examples like $$$x \in$$$ {$$$2,3$$$}, $$$y=5$$$. But I don't quite get the logic behind that and I'm not sure if it's a general rule that $$$kx \leq y \rightarrow x \leq \lfloor y/k \rfloor$$$.

Any help with understanding that or resources to read about it would be greatly appreciated. Thanks!

Auto comment: topic has been updated by SuperSheep (previous revision, new revision, compare).Just read the definition of ceil and floor.

it's just simple math, im assuming all the variables here are int, therefore automatically rounded down, when you divide a by b, and a % b != 0, the result gets rounded down to the integer part

If $$$kx\leq y$$$, then $$$x\leq\frac yk$$$. By the definition of floor, the largest integer less than or equal to $$$\frac yk$$$ is $$$\left\lfloor\frac yk\right\rfloor$$$ so if $$$x$$$ is also an integer then $$$x\leq\left\lfloor\frac yk\right\rfloor$$$.