An Illustration of Number Theory with Groups

I will start off by introducing a few important definitions that deal with relevant concepts and principles.

Integer - Meaning “untouched” or “whole” in Latin, and coming from Zahlen, the German word for number, the integers are the set of numbers {…, -3, -2, -1, 0, 1, 2, 3, …}. In Mathematics, the integers are commonly denoted as Z.
Group – An algebraic structure consisting of a set combined with a binary operation that satisfies the group axioms (laws): closure, associativity, identity, and inverses.

Without loss of generality, we will ignore the meaning of the group axioms, as they are not essential for sufficient comprehension of this particular example.
Now that the boring stuff is out of the way, we can get to the real meat. The main idea here is to establish an understanding of congruence classes, or more accurately, congruence relations.
Congruency takes advantage of a useful form of calculation that is called Modular Arithmetic. If you haven’t heard of modular arithmetic, you are probably in the same boat as most, however, that does not mean you haven’t been exposed to it on a daily basis. It may sound crazy, but the simplest example is a clock’s representation of time.
Think of 12 as 0. Why, because a clock works modulo 12. This essentially means that any number divisible by 12 is equivalently 0, hence, the congruence relation. I will illustrate this example mathematically:Z/12 = {0,1,2,3,4,5,6,7,8,9,10,11}
Where the / denotes modulo, i.e. any integer is taken modulo 12 where multiples of 12 are discarded and all that remains is a natural number between 0 and 11.

Z/12 forms a group, in particular, an Abelian group, but this is just more fancy math talk for a commutative group. Why is this group commutative? Its binary operation is addition, and the addition of any two or more numbers is commutative. Sometimes, the group Z/12 with addition as the operation is represented as (Z/12, +). Notice that also forms a group (Z/12, x). Explaining this is more complicated, therefore I will get back to the topic on hand.
What has happened with the clock is that we have taken the “entire” set of integers and formed an equivalence class, that is to say, every integer is congruent to either 0,1,2,3,4,5,6,7,8,9,10, or 11.

Thank God right?? If we didn’t take advantage of this, and the start of time were marked 13.6 billion years ago, the time Scientists believe the Big Bang occurred, knowing that there are roughly 8,760 hours each year, if somebody currently asked you what time it is, you would have to respond approximately 1.19136 x 10^14 hours.

What a waste of time!