Importance of number properties & inequalities

Getting problems on number properties correct is verrrrry important to get a score above 49 on quants. During my practice, I have observed that maximum tricks can be incorporated in questions from number properties and the human mind (specially mine) tends to forget to take care of those small yet significant tricks under time constraints. To reinforce in my brain and in all of your brains, always remember the following while doing number properties and inequalities:

  1. ALWAYS consider zero a value of variable. For example, If a*b*a = a, we CANNOT simply divide both sides by a to get ab=1 because a could be zero and we can not divide by zero. Rather it should be worked as: a*b*a-a=0. It implies a(ab-1) =0 —> either a=0 or ab =1.
  2. NEVER assume that any variable is integer unless clearly mentioned.
  3. You can only ADD inequalities when their signs are in the SAME direction. If a>b and c>d, then (a+c )> (b+d)
  4. You can only SUBTRACT inequalities when their signs are in the OPPOSITE direction. If a>b and c <d, then (a-c) > (b-d). Take the sign of inequality you subtract from.
  5. Whenever there is a square of a variable involved, ALWAYS consider that the variable may be an irrational number which becomes integer when its squared. For example, if its given that n^2 is an integer, it does not necessarily mean that n will be an integer. n could be 2^1/2 as well.
  6. In modulus problems, ALWAYS put back the values derived in the equation to validate.
  7. In inequalities, NEVER divide or multiply both sides of an equation with unknown variable (whose sign is not known) because in case the variable has a negative value, the sign of inequality shifts.
I will update this list as and when I make more errors and learn lessons from them. I hope this list will be of help to someone out there.