ABSTRACT
When you classify a number as even or odd, you’re using the concept of congruence. In the Division Algorithm, an even number is a number that leaves a remainder of 0 when divided by 2 and an odd number leaves a remainder of 1 when divided by 2. Although nobody would say that 14 and 96 are the same number, they do share the property of having the same remainder when divided by 2. Similarly, when you look at a clock and you notice that it’s 10:00 now and think to yourself that in 5 hours it will be 3:00, you’re not saying that 10 + 5 = 3. What you’re really doing, perhaps without realizing it, is saying 10 + 5 = 15, and when I divide 15 by 12, the remainder is 3. Congruences generalize these ideas to all positive integers by saying (as we’ll see in a bit) that two numbers are “congruent for the number n” if they leave the same remainder when divided by n. So, to return to our clock example, 3 and 15 “are congruent for the number 12” because they both leave a remainder of 3 when divided by 12. It will take us a few paragraphs to make this precise and we’ll have to develop some notation along the way, so let’s begin.
