Modulo Operation/Examples/5 mod 3

Theorem

$5 \bmod 3 = 2$

where $\bmod$ denotes the modulo operation.

Proof

By definition of modulo operation:

$x \bmod y := x - y \floor {\dfrac x y}$

for $y \ne 0$.

We have:

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

and so:

$\floor {\dfrac 5 3} = 1$

Thus:

$5 \bmod 3 = 5 - 3 \times \floor {\dfrac 5 3} = 5 - 3 \times 1 = 2$

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: $(3)$