Modulo Operation/Examples/18 mod 3

Theorem

$18 \bmod 3 = 0$

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:

$\dfrac {18} 3 = 6 + \dfrac 0 3$

and so:

$\floor {\dfrac {18} 3} = 6$

Thus:

$18 \bmod 3 = 18 - 3 \times \floor {\dfrac {18} 3} = 18 - 3 \times 6 = 0$

$\blacksquare$

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: $(3)$