Christopher wrote:
anytime you can rearrange an equation to get a 10 or 100 or the like in there, it makes it easier to solve.

This is very true. There are lots of "tricks" you can use  like knowing the digits of anything divisible by 3 will, themselves, sum up to something divisible by 3.
For example:
1386486
Sum the digits: 36.
Sum the digits again: 9.
So it's divisible by 3.
Similarly anything divisible by 9 can have its digits added up to something divisible by 9. (So 1386486 above is divisible by 9.)
You learn them mostly through experience, but also sometimes through being taught them.