Modulo Operation/Examples/100 mod 3
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Theorem
- $100 \bmod 3 = 1$
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:
- $\dfrac {100} 3 = 33 + \dfrac 1 3$
and so:
- $\floor {\dfrac {100} 3} = 33$
Thus:
\(\ds 100 \bmod 3\) | \(=\) | \(\ds 100 - 3 \times \floor {\dfrac {100} 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 100 - 3 \times 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
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: Exercise $8$