Modulo Operation/Examples/-2 mod -3

Theorem

$-2 \bmod -3 = -2$

where $\bmod$ denotes the modulo operation.

Proof

By definition of modulo operation:

$x \bmod y := x - y \floor {\dfrac x y}$

for $y \ne 0$.

We have:

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

and so:

$\floor {\dfrac {-2} {-3} } = 0$

Thus:

\(\ds -2 \bmod -3\)

\(=\)

\(\ds -2 - \paren {-3} \times \floor {\dfrac {-2} {-3} }\)

\(\ds \)

\(=\)

\(\ds -2 - \paren {-3} \times 0\)

\(\ds \)

\(=\)

\(\ds -2\)

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $9$