Modulo Operation/Examples/100 mod 7

Theorem

$100 \bmod 7 = 2$

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:

$\dfrac {100} 7 = 14 + \dfrac 2 7$

and so:

$\left\lfloor{\dfrac {100} 7}\right\rfloor = 14$

Thus:

 $\displaystyle 100 \bmod 7$ $=$ $\displaystyle 100 - 7 \times \left\lfloor{\dfrac {100} 7}\right\rfloor$ $\displaystyle$ $=$ $\displaystyle 100 - 7 \times 14$ $\displaystyle$ $=$ $\displaystyle 2$

$\blacksquare$