Modulo Operation/Examples/18 mod 3

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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:

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

$\blacksquare$


Sources