Modulo Operation/Examples/5 mod -3

Theorem

$5 \bmod -3 = -1$

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:

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

and so:

$\floor {\dfrac 5 {-3} } = -2$

Thus:

\(\ds 5 \bmod -3\)

\(=\)

\(\ds 5 - \paren {-3} \times \floor {\dfrac 5 {-3} }\)

\(\ds \)

\(=\)

\(\ds 5 - \paren {-3} \times \paren {-2}\)

\(\ds \)

\(=\)

\(\ds 5 - 3 \times 2\)

\(\ds \)

\(=\)

\(\ds -1\)

$\blacksquare$

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 $9$