Modulo Operation/Examples/5 mod 3

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