Modulo Operation/Examples/-100 mod 7

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: