Modulo Operation/Examples/-2 mod 3

Theorem

 * $-2 \bmod 3 = 1$

where $\bmod$ denotes the modulo operation.

Proof
By definition of modulo operation:
 * $x \bmod y := x - y \floor {\dfrac x y}$

for $y \ne 0$.

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

and so:
 * $\floor {\dfrac {-2} 3} = -1$

Thus:
 * $-2 \bmod 3 = -2 - 3 \times \floor {\dfrac {-2} 3} = -2 - 3 \times \paren {-1} = 1$