Floor of x+m over n/Corollary

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Corollary to Floor of $\frac {x + m} n$

Let $n \in \Z$ such that $n > 0$.

Let $x \in \R$.


Then:

$\left \lfloor{\dfrac x n}\right \rfloor = \left \lfloor{\dfrac {\left \lfloor{x}\right \rfloor} n}\right \rfloor$

where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.


Proof

This is a special case of Floor of $\dfrac {x + m} n$:

$\left \lfloor{\dfrac {x + m} n}\right \rfloor = \left \lfloor{\dfrac {\left \lfloor{x}\right \rfloor + m} n}\right \rfloor$

where $m = 0$.


Sources