Modulo Operation/Examples/5 mod -3

Theorem

 * $5 \bmod -3 = 5$

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:
 * $\dfrac 5 {-3} = -2 + \dfrac 1 3$

and so:
 * $\left\lfloor{\dfrac 5 {-3} }\right\rfloor = -2$

Thus: