Proving statements using Euclidean division I have a series of statements that are proved based on the equation for Euclidean division, this is: 

Given two integers $a$ and $b$, with $b ≠ 0$, there exist unique
  integers $q$ and $r$ such that $a = bq + r$ and $0 ≤ r < |b|$, where
  $|b|$ denotes the absolute value of $b$.

And these are the statements:
1) The square value of every odd integer can be written as $8k+1$.
2) If $p ≠ 3$ is a prime number then $p²+2$ is not a prime number.
3) If $a$ is an integer number that can not be divided by $2$ and $3$, then $24$ divides $(a²-1)$.
4) The sum of the square values of two odd integer numbers can not be a square value.
5) Prove that if $p \geq q \geq 5 $ and $p$ and $q$ are prime numbers then $24|(p²-q²)$.
And this are the initial values given to $b$ in the Euclidean equation for each statement in order to prove the statements:
1) $4q+r$
2) $6q+r$
3) $12q+r$
4) $4q+r$
5) $12q+r$
My question is: How is the value of $b$ assigned?
 A: 
It is better to start investigating the problem you want, by choosing
  $b$ to be the largest possible,or the smallest possible.

My answer is: For proving that some number $k$ divides an expression start by searching modulo $k$.(Note that (1) is actually saying that $8|a^2-1$ is true if $a$ is odd.
For proving that a number is composite or something general like statement (4),start from the bottom: search modulo $2,3,4,5...$
For example for the first statement it is better to assume that the odd integer has the form $8q+1,8q+3,8q+5,8q+7$ rather than start with $2q+1$.
For the last statement it is better to consider the cases modulo $24$ for the prime numbers.
Every prime $p$ for example is written only in one of the forms $24q+1,24q+5,24q+7,24q+11,24q+13,24q+17,24q+19,24q+23$ (Note that modulo $12$ you have $4$ cases $1,5,7,11$ mod $12$ but you need to do some maths  in order to prove what you want).
So, if you want to start from somewhere without thinking it too much,start from the biggest moduli.
If you want to prove that a number of some form is not prime start searching from the least one. Try division with $2,3,4,5...$ sooner or later you will see something bright coming out of the fog.     
The same happens when you want to prove case 4)
 You will see that modulo $2$ or $3$ nothing is clear but you are only a step away:modulo $4$.  
