So given a set $S = a, a^2, a^3, ldots$, with relation $a | b leftrightarrow a leq b$, does the relation hold going from left to right or right to left? i.e. $a|a^2, a^2|a^3,ldots$
$egingroup$ Let $(G,circ)$ be a group. We then say that $amid b$ (read as "$a$ divides $b$") if, $$exists cin Gmid b=acirc c$$ $endgroup$
Given two integers $a$ and $b$, we say $a$ divides $b$ if there is an integer $c$ such that $b=ac$. Source.
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This is what $a$ divides $b$ means. The shorthand notation is$$a|b$$.
In your example, $$a|a^2iff aleq a^2$$since by definition there exists $c$ such that $a^2 = ac$, namely $a = c$.
We say $a$ divides $b$, denoted by $a | b$, if $b$ is a multiple of $a$ (ie, $b$ is an integer multiple of $a$). Equivalently, $a |b$ iff $b=ka$ for some integer $k$.
To remember what "$2$ divides $6$" means, perhaps you can remember the phrase "$2$ divides $6$ into $3$ parts". Hence, $2 | 6$.
Note that $2 | 0$ because $0$ is an integer multiple of $2$: $0 = k2$ for some integer $k$. Just take $k=0$.
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Prove the relation 'x divides y' on the natural numbers is antisymmetric but not on the integers.
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