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