The purpose of this section is twofold. First, Now that we have some experience with mathematical proof, we’re now going to expand the types of questions we can prove by introducing the Divides and Congruence relations. Second, this is the first step in building the tools we need towards working with some encryption algorithms.
SubsectionThe Divides Relation
Note3.1.1.
Any time we say “number” in the context of divides, congruence, or number theory we mean integer.
In Example 1.3.3, we saw the divides relation. Because we’re going to use this relation frequently, we will introduce its own notation.
Definition3.1.2.The Divides Relation.
Let \(a \and b\) be two integers with \(a \not= 0\text{.}\) We say \(a\) divides \(b\) and write \(a \divides b\) if there exists an integer \(m\) such that \(b = am\text{.}\)
We say that \(a\) is a factor of \(b\text{,}\) and \(b\) is a multiple of \(a\text{.}\)
Example3.1.3.
The following are examples of the divides relation:
\(3 \divides 6\) since \(6 = 3\cdot 2\)
\(4 \divides 100\) since \(100 = 4 \cdot 25\)
Here are some non-examples:
\(4 \nmid 10\) since there is no integer \(m\) for which \(10 = 4m\text{.}\)
\(6 \nmid 3 \text{.}\) Order matters.
This is a good time to remember that the relational statment \(a \mid b\) is a propositional statement. It is true or false. Let’s compare the divides relation with some similar symbols.
Example3.1.4.
What kind of object are each of the following? What is the value, truth or numeric?
If \(a, b, c\in \Z\) with \(a\not= 0\) such that \(a\divides b\) and \(a \divides c\text{,}\) then \({ a\divides (bm+cn)}\) where \(m \and n\) are some integers.
Let’s compare the the congruence relation with the modulus operator:
\(a \equiv b \pmod{m}\) is a logical statement. It tells us that \(a\) and \(b\) are related, that is, the statement \(m \divides (b-a)\) is true.
\(a \modulus b\) is a mathematical operation. The result is a number.
The difference is subtle but important.
Definition3.1.12.Congruence Classes.
If \(a\) is an integer and \(m\) is a positive integer, we define the congruence class of \(a\) modulo \(m\) to be the set of all integers congruent to \(a\) modulo \(m\text{.}\)
\begin{equation*}
\left[ a \right]_m = \left\{ b \mid b \equiv a\pmod{m} \right\}
\end{equation*}
In general, there will always be \(m\) distinct congruence classes modulo \(m\text{.}\)
Proposition3.1.15.
Let \(m\) be a positive integer. Then integers \(a \and b\) are congruent modulo \(m\) if and only if there exists and integer \(k\) such that \({a = b + km}\text{.}\)
We call the collection of the operations of addition and multiplication modulo \(m\) modular arithmetic.
Note3.1.17.
For modular arithmetic, we have addition and multiplication. Subtraction is done via additive inverses, but division is not a defined operation! We’ll discuss multiplicative inverses in section 3.4.
ExercisesExercises
1.
Show that if \(a \equiv b\pmod{m}\text{,}\) then \(b \equiv a \pmod{m}\)
Assume that \(a, b, c\) are integers with \(a \ne 0\) and \(c\ne 0\) such that \(ac \divides bc\text{.}\) Then there exists an integer \(m\) such \(bc = acm\)
Since \(c \ne 0\text{,}\) we can divide both sides of the equation by \(c\text{,}\) yielding the equality \(b = am\text{.}\) Thus \(a \divides b\text{.}\)
3.
Use the division algorithm to write the following divisions as \(n = dq + r\) with the variables defined in the theorem:
These numbers are \(-1, -26, -51, -76, 24, 49, 74, 99,\) all numbers that are of the form \(-1 + 25 \cdot k\) for integers \(k\text{.}\)
6.
Suppose that \(a \and b\) are integers, \(a \equiv 4 \pmod{13}\) and \(b \equiv 9\pmod{13}\text{.}\) Find the integer \(c\) with \(0 \le c \le 12\) such that:
One possible counterexample is \(a = 2, b=4, c=4, and m=6\text{.}\) Certainly \(2\cdot 3 \equiv 4 \cdot 3 \pmod 6\) but \(2 \not \equiv 4 \pmod 6\text{.}\)
One possible counterexample here is \(a = 2, b=7, c=7, b=2, m=5\text{.}\) Then we can see \(2\equiv 7\pmod 5\) satisfies both parts of the hypothesis, but \(2^7 \equiv 3 \pmod 5\) while \(7^2 \equiv 4 \pmod 5\text{.}\)