1. The most difficult part of the reading was grasping the idea of division with congruences. The actual act of dividing makes sense, the only trouble is to remember that you can divide both sides of the congruence sign only if a (the divisor) and n (the modulo) are relatively prime. This I must remember. I also must remember that if the gcd of a and an is greater than 1, I have to step back and examine a few things? If gcd(a,n) does not divide b, then there is no solution. If d does divide b then you can divide a and b by d but you must remember to divide n by d as well. Addition, subtraction, and multiplication all work just fine in congruences, it's division that I must remember. Even more so, fractions seem to be a hairy deal to mess with when working with congruences.
2. I think about congruences and how they relate to equalities. You can do some similar operations (as mentioned above) on both. It seems as if congruences modulo n package larger numbers down. I've taken Abstract Algebra, but we haven't really dealt with a whole lot of applications. I'm not a huge fan of applications, but I would be interested in pursuing a career as some sort of information analyst. I recognize that by using congruences, you can simplify a lot of information in such a way so as to make it easier to spot patterns and the like. I don't know exactly how, though.
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