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:
- $x \bmod y := x - y \floor {\dfrac x y}$
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: $(3)$