I looked at a few modular arithmetic websites and I'm a neophyte when it comes to the legal operations/rules, syntax, and procedures of modular arithmetic. No GRE guide I've seen talks about modular arithmetic (not even the official one!).

Following are five relevant questions from my GRE coaching website (which only gives answers with no explanation).

Trial-and-error is always an option but I'm looking for a systematic approach, clear, efficient step-by-step procedure, which gives me confidence to tackle such problems.
How do I set up the congruent modulo expression?
What operations are allowed in that expression, and which one's are usually performed to reach answers to the below questions with this approach?

number theory. study it every day for a year and it still wont be enough for the GRE.
1. ? too many variables.
2. 10=-1 mod 11, 100=-1 mod 11...etc
3. 3+3^2=12 =0 mod 6. error.
4. set them equal. inspection.
5. 1/3 mod (1) = ?
6. by inspection. start with small numbers.
7. i didn't read.

notes:
I. for all x>y integrers there exist M,R<y such that x=y*M +R
II. for R=0 ,y divides x or x is a multiple of y; R>0, y divides x-R
III. x = R mod y is the same as y divides x-R which is the same as x=y*M+R