Divisibility. For $a,$binZ,$a0,$ we say a divides $b$ if there exists $($a unique$)$ an integer $k$ with $b=ak.$

Example: $3|-12||,$ since $-12=3*4,$ but $124.$

Notes:

Closure of $Z$ under multiplication $($and other things$)$ plays a big role in these arguments, and can be

assumed.

More or less, throughout this and the next few batches, all of our numbers will be integers. So$,$ if you

fail to say kinZ it's okay. But closure should be cited when used.

The definition we used in class for divides included the "unique." Though you may have noticed one of

a couple things: $1.$ The definition above adds it as a parenthetical. $2.$ None of our arguments showing

$a|b||$ or $a|||$ anything ever worried about the 'unique' part of the statement $($nor should they$).$ Well,

that is because, as a student pointed out, the uniqueness follows from the definition. Show this. That

is$,$ show that the $k$ in the above definition for $a|b||$ is unique. $($Note: your argument needs to be over

$Z.$ No quotients.$)$