logic - Properties of modular arithmetic

I've recently learned about some modular congruence, but I'm having trouble actually solving problems. I've tried a few references--http://www.math.cornell.edu/~putnam/modular.pdf gets pretty close, but I'm still a bit confused.

For example, let's say I have $$6a\equiv 10b \mod{14}$$

What else can I say is true? Can I just divide everything by a factor? For example, is $$3a \equiv 5b \mod{7}$$

By experimentation, I think the above is true.

So then, I can just divide everything by the same number. But, in the example below (which I think is true?) I divide x and y by 26, but I don't change the mod $$26x \equiv 26y \mod{5}$$ and therefore $$x \equiv y \mod{5}$$

Answer

Yes, the two things are both true.

The first one is:

$$ad \equiv bd \pmod{nd}$$
if and only if
$$a \equiv d \pmod{n}$$

This happens because
$$dn |d(a-b) \Leftrightarrow n |a-b$$

The second is a completely different fact, which happens for a different reason:

If
$$ad \equiv bd \pmod{n } \mbox{ and } gcd(d,n)=1 \mbox{then} \\ a \equiv b \pmod{n}$$

This is because $n|d(a-b)$ and $gcd(n,d)=1$ implies $n|a-b$.