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Section 2.1 Integers and divisibility

The symbol \(\Z\) (from a German word for number) represents the \dfn{integers}, the space of positive and negative whole numbers.

\begin{equation*} \Z = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} \end{equation*}

Number theory is the area of mathematics concerned with the properties and behavior of the numbers in . \(\Z\) is an interesting object of study because of the structures that it contains.

First, let's lay out some basic definitions. The additive identity in \(\Z\) is the number \(0\text{,}\) since \(a + 0 = a\text{.}\) The multiplicative identity in \(\Z\) is \(1\text{,}\) since \(a \cdot 1 = a\text{.}\) The integers have the following structural properties:

  1. \(\Z\) is closed under addition: if \(a, b \in \Z\text{,}\) then \(a + b \in \Z\text{.}\)
  2. \(\Z\) is closed under multiplication: if \(a, b \in \Z\text{,}\) then \(ab \in Z\text{.}\)
  3. Every integer has an additive inverse: if \(a \in \Z\text{,}\) then there exists a \(b \in \Z\) so that \(a + b = 0\text{.}\)

Essentially, this means that addition, subtraction, and multiplication are nicely defined, completely consistent operations in \(\Z\text{.}\) But, almost no elements of \(\Z\) have a multiplicative inverse in \(\Z\): that is, for a given \(a \in \Z\text{,}\) it is almost never the case that there is a \(b \in \Z\) so that \(ab = 1\text{.}\)

This essentially means that division is an operation that sometimes is not well-defined in \(\Z\) and related sets of numbers. Quite a bit of interesting and useful math involves the study of when division is possible.

Definition 2.1.1.

Let \(a, b \in \Z\text{.}\) Say that “\(b\) divides \(a\)” or “\(a\) is divisible by \(b\)” if there exists an integer \(q \in \Z\text{,}\) called the quotient, so that

\begin{equation*} a = bq. \end{equation*}

When \(b\) divides \(a\text{,}\) we write \(b\mid a\text{.}\) When \(b\) does not divide \(a\text{,}\) we write \(b \nmid a\text{.}\)

Divisibility has several simple but extremely useful properties.

We close this section with a seemingly basic but extremely important definition usually called the division algorithm. The division algorithm gives a decomposition of any integer \(a\) after division by \(b\text{,}\) even when \(a\) and \(b\) do not have a divisibility relationship.

Definition 2.1.3. (Division algorithm).

Let \(a, b \in \Z\text{.}\) Then there exist a unique integer \(q\) and a unique positive integer \(r\) with \(0 \leq r \lt b\) so that

\begin{equation*} a = bq + r. \end{equation*}

\(q\) is called the quotient and \(r\) is called the remainder.

The python function divmod(a,b) will find quotients and remainders, though we're going to look at implementing our own functions in the sequel section on the Euclidean algorithm.

Exercises Exercises

1.
\(\Z\)\(1\)\(-1\)
2.
\(\Z_{12}\)