1. ## Modulo and congruence

This one seems to easy.

if $\displaystyle a\not\equiv b \ \mbox{(mod m)}$, then $\displaystyle m\nmid (a-b)$

Does it suffice to just do this $\displaystyle a\not\equiv b \ \mbox{(mod m)}\rightarrow m\nmid (a-b)$ and now I am done?

2. Originally Posted by dwsmith
This one seems to easy.

if $\displaystyle a\not\equiv b \ \mbox{(mod m)}$, then $\displaystyle m\nmid (a-b)$

Does it suffice to just do this $\displaystyle a\not\equiv b \ \mbox{(mod m)}\rightarrow m\nmid (a-b)$ and now I am done?

$\displaystyle a\not\equiv b \ \mbox{(mod m)}$, then$\displaystyle \frac{a-b}{m} \neq k\in \mathbb{Z}$
So it is clear that no such $\displaystyle k$ exist, hence, $\displaystyle m\nmid (a-b)$

3. Perhaps you could insert the intermediate step using the definition of congruence:

$\displaystyle a\not\equiv b\;(\!\!\!\!\mod m)$ implies $\displaystyle \neg(\exists n\in\mathbb{Z})(mn=a-b)$ implies $\displaystyle m\not|(a-b)$.

[EDIT] Which is the same thing as AsZ's post.

4. So it is just that easy than, thanks.