I have my very first mathematical reasoning exam tomorrow, and I'm extremely worried because our professor literally doesn't teach nor tell us anything since the class is mostly "group" work. I'm also in a Discrete Mathematics class and we prove statements by one line at a time with a reasoning at the right hand side. Should I do the same for this exam?
For example: Let a and b be integers and n a natural number
CLAIM: $$ a \equiv b \ mod \ n \Rightarrow a^2 \equiv b^2 \ mod \ n $$
PROOF:
$$ n|(a-b) \ \ by \ definition \ of \ congruency \ and \ modulo $$ $$ (a-b) = n*q \ \ for \ some \ integer \ q \ by \ definition \ of \ divisibility $$ $$a = n*q + b$$ $$a^2 = (nq + b)^2 = (n^2q^2 + 2bnq + b^2) \ \ what \ reasoning \ can \ I \ use \ here?$$ $$ a^2 - b^2 = n^2q^2 + bnq + b^2 - b^2$$ $$ a^2-b^2 = n(nq^2 + bq ) $$ $$a^2 - b^2 = n (c) \ \ for \ some \ integer \ c $$ $$ n | (a^2 - b^2) $$ $$ Therefore, \ a^2 \equiv b^2 \ mod \ n $$
There are some in which I don't know which reasoning I would put next to each line. One thing that worries me about this proof is that I am combining a product of an integer and a natural number $(nq^2 + bq)$ and saying that they are an integer $c$. Can we assume that?
I also apologize that the proof is not in a great format.
