Modulo Operation/Examples/-2 mod 3

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Theorem

$-2 \bmod 3 = 1$

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:

$\dfrac {-2} 3 = -1 + \dfrac 1 3$

and so:

$\left\lfloor{\dfrac {-2} 3}\right\rfloor = -1$

Thus:

$-2 \bmod 3 = -2 - 3 \times \left\lfloor{\dfrac {-2} 3}\right\rfloor = -2 - 3 \times \left({-1}\right) = 1$

$\blacksquare$


Sources