# Existence of q for which j - qk is Positive

## Theorem

Let $j, k \in \Z$ be integers such that $k > 0$.

Then there exist $q \in \Z$ such that $j - q k > 0$.

## Proof

Let $q = -\size j - 1$.

Then:

 $\displaystyle j - q k$ $=$ $\displaystyle j - \paren {-\size j - 1} k$ $\displaystyle$ $=$ $\displaystyle j + \size j + k$

We have that:

$\forall j \le 0: j + \size j = 0$

and:

$\forall j > 0: j + \size j = 2 j$

So:

$j - q k \ge k$

and as $k > 0$ the result follows.

$\blacksquare$