Modulo Operation/Examples/18 mod 3

Theorem

 * $18 \bmod 3 = 0$

where $\bmod$ denotes the modulo operation.

Proof
By definition of modulo operation:
 * $x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$

for $y \ne 0$.

We have:
 * $\dfrac {18} 3 = 6 + \dfrac 0 3$

and so:
 * $\left\lfloor{\dfrac {18} 3}\right\rfloor = 6$

Thus:
 * $18 \bmod 3 = 18 - 3 \times \left\lfloor{\dfrac {18} 3}\right\rfloor = 18 - 3 \times 6 = 0$