I'm a very newbie, even if I studied some math at college... Now I'm trying to study "Contemporary Abstract Algebra", by J.A. Gallian but I'm already facing some difficulties trying to understand the proof of the very first theorem (0.1) about the division algorithm.
The theorem says:
Let a and b be integers with $b > 0$. Then there exist unique integers q and r with the property that $a=bq + r$, where $0\le r\le b$
Trying to prove the existence part of the theorem, the author writes:
Consider the set $S =\{a - bk |\ k\text{ is an integer and }a - bk\ge 0\}$. If $0\in S$, then $b$ divides $a$ and we may obtain the desired result with $q =\tfrac ab$ and $r = 0$. Now assume $0\notin S$. Since $S$ is nonempty [if $a > 0$ then $a - b\times0\in S$ and if $a < 0$ then $a - b(2a) = a(1 - 2b) \in S$; and $a\neq0$ since $0\notin S$]...
Ok, my question is: why author put $k = 2a$ when explain why S is non empty with $a < 0$? Where does this $k = 2a$ come from? Why can't be $k=a$, for example?
Thanks a lot
The author chose $k=2a$ because for the most obvious choice $k=a$ there is a problem: If $b=1$ then $a-b\times a=0$, which is not in $S$.
For $k=2a$ we have $a-b\times2a=a(1-2b)\in S$ because certainly $1-2b<0$. So this does show that $S$ is nonempty.