Existence of q for which j - qk is Positive

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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.

$\blacksquare$


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