# Modulo Operation/Examples/-100 mod 7

## Theorem

$-100 \bmod 7 = 5$

where $\bmod$ denotes the modulo operation.

## Proof

By definition of modulo operation:

$x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor$

for $y \ne 0$.

We have:

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

and so:

$\left\lfloor{\dfrac {-100} 7}\right\rfloor = -15$

Thus:

 $\displaystyle -100 \bmod 7$ $=$ $\displaystyle -100 - 7 \times \left\lfloor{\dfrac {-100} 7}\right\rfloor$ $\displaystyle$ $=$ $\displaystyle -100 + 7 \times 15$ $\displaystyle$ $=$ $\displaystyle 5$

$\blacksquare$