Algebra and Number Theory/Elementary Number Theory

=Divisibility= Definition 1: (divides, divisor, multiple)

Let $$a, b \in \mathbb{Z}$$, with $$a \neq 0$$. We say that "$$a$$ divides $$b$$" or that "$$b$$ is a multiple of $$a$$", if there exists some $$q \in \mathbb{Z}$$ such that $$b = aq$$.

We write this as $$a \mid b$$.

Proposition 1: (some elementary properties of division)

Let $$ a, \ b, \ c, \ n, \ m $$ be integers. Then Examples: $$ 3|6 $$ because $$ 6 = 3 \times 2 $$. However $$ 3\nmid 7 $$: if it did, would also divide $$ 1 $$ (by Proposition 1, point 3), which is impossible (Proposition 1, point 1). Similarly, $$ 3\nmid 8 $$.
 * 1) If $$ a, \ b > 0 $$ and $$a \mid b$$, then $$ a \leqslant b $$.      ▶ $$ b = |b| = |aq| = a|q| \geqslant a, \ q \in \mathbb{Z}_{>0} $$ □
 * 2) If $$a \mid b$$ and $$b \mid a$$, then $$a = \pm b$$.
 * 3) If $$a \mid b$$ and $$a \mid c$$, then $$a \mid nb + mc$$.
 * 4) If $$a \mid b$$ and $$b \mid c$$, then $$a \mid c$$.                ▶ $$ c = qb = q(ra) = (qr)a $$ □

Proposition 2: (division algorithm)

Let $$a, b \in \mathbb{Z}$$, with $$a \neq 0$$. Then $$ b = aq + r $$, for some $$ q, \ r \in \mathbb{Z} $$, with $$ 0 \leqslant r < a $$.

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