Modulo Operation/Examples/18 mod -3

Theorem

 * $18 \bmod -3 = 0$

where $\bmod$ denotes the modulo operation.

Proof
By definition of modulo operation:
 * $x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$

for $y \ne 0$.

We have:
 * $\dfrac {18} {-3} = -6 + \dfrac 0 3$

and so:
 * $\left\lfloor{\dfrac {18} {-3} }\right\rfloor = -6$

Thus: