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