Modulo Operation/Examples/-100 mod 7

Theorem

$-100 \bmod 7 = 5$

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:

$\dfrac {-100} 7 = -15 + \dfrac 5 7$

and so:

$\floor {\dfrac {-100} 7} = -15$

Thus:

\(\ds -100 \bmod 7\)

\(=\)

\(\ds -100 - 7 \times \floor {\dfrac {-100} 7}\)

\(\ds \)

\(=\)

\(\ds -100 + 7 \times 15\)

\(\ds \)

\(=\)

\(\ds 5\)

$\blacksquare$

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 $8$