Modulo Operation/Examples/5 mod 3

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Theorem

$5 \bmod 3 = 2$

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:

$\dfrac 5 3 = 1 + \dfrac 2 3$

and so:

$\left\lfloor{\dfrac 5 3}\right\rfloor = 1$

Thus:

$5 \bmod 3 = 5 - 3 \times \left\lfloor{\dfrac 5 3}\right\rfloor = 5 - 3 \times 1 = 2$

$\blacksquare$


Sources