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:
\(\ds j - q k\) | \(=\) | \(\ds j - \paren {-\size j - 1} k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds j + \size j + k\) |
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.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-1}$ Euclid's Division Lemma: Exercise $1$