Modulo Operation/Examples/-100 mod 7

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Theorem

$-100 \bmod 7 = 5$

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:

$\dfrac {-100} 7 = -15 + \dfrac 5 7$

and so:

$\floor {\dfrac {-100} 7} = -15$


Thus:

\(\ds -100 \bmod 7\) \(=\) \(\ds -100 - 7 \times \floor {\dfrac {-100} 7}\)
\(\ds \) \(=\) \(\ds -100 + 7 \times 15\)
\(\ds \) \(=\) \(\ds 5\)

$\blacksquare$


Sources