Modulo Operation/Examples/-2 mod -3

Theorem

 * $-2 \bmod -3 = -2$

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:
 * $\dfrac {-2} {-3} = 0 + \dfrac 2 3$

and so:
 * $\left\lfloor{\dfrac {-2} {-3} }\right\rfloor = 0$

Thus: