3.7 Modular arithmetic

There are an infinite number of integers and hence rationals and real numbers. Are there sets of numbers that behave somewhat like them but are finite?

Consider the integers modulo 6: {0, 1, 2, 3, 4, 5}. We write 3 mod 6 when we consider the 3 in this collection. Given any integer n, we can map it into this collection by computing the remainder modulo 6. Arithmetic can be done in the same way:

Instead of using ‘‘=’’, we write ‘‘≡’’ and say that a is congruent to b mod 6 when we see a ≡ b mod 6. This means that a − b is evenly divisible by 6: 6 | (a − b). This is a group under addition with identity 0. In the last example, 2 is the additive inverse of 4. We denote this Z/6Z.

Question 3.7.1

What is −1 mod 6? For n a natural number greater than 1, what is −1 mod n?

Let’s consider the same...