4x mod 5 = 3x-1 mod 5
Should I do:
5|(4x-(3x-1))
or
5|((3x-1)-4x)?
according to the theorem, it should be: if a mod 5 = b mod 5
then 5|(a-b), but shouldn't we take the greatest number?
4x mod 5 = 3x-1 mod 5
Should I do:
5|(4x-(3x-1))
or
5|((3x-1)-4x)?
according to the theorem, it should be: if a mod 5 = b mod 5
then 5|(a-b), but shouldn't we take the greatest number?
It makes no difference at all. Suppose that $m\mid a-b$; then there is an integer $k$ such that $a-b=km$. But then $b-a=-(a-b)=-km$, and $-k$ is an integer, so $m\mid b-a$. Thus, the following four statements are all equivalent:
This is good to know:
$a\equiv b\pmod m \;\iff \;m\mid a-b \;\iff \; m\mid (b-a) \;\iff \; b\equiv a\pmod m \quad\quad\quad\tag{1}$
$ \text{and }\;\; m\mid c \quad\iff \quad m\mid -c \tag{2}$
So what you've written is true, both statements:
$5\mid (4x-(3x-1))\quad \text{AND}\quad 5\mid((3x-1)-4x)$ That is: $5\mid (x+1)\quad \iff \quad 5\mid -(x + 1)$
So $x \equiv -1 \pmod 5\quad\iff \quad x\equiv 4 \pmod 5$.
Indeed, any $x = 5k + 4$ where $k$ is any integer is a solution: $x \in \{ \cdots, -6, -1, 4, 9, ...\}$.