Skip to main content

Beginning Activity Beginning Activity 2: Some Work with Congruence Modulo \(\boldsymbol{n}\)
A4US

1.

Let \(n\) be a natural number and let \(a\) and \(b\) be integers.

(a)

Write the definition of ā€œ\(a\) is congruent to \(b\) modulo \(n\text{,}\)ā€ which is written \(a \equiv b \pmod n\text{.}\)

(b)

Use the definition of ā€œdividesā€ to complete the following:

When we write \(a \equiv b \pmod n\text{,}\) we may conclude that there exists an integer \(k\) such that ā€¦.

We will now explore what happens when we multiply several pairs of integers where the first one is congruent to 3 modulo 6 and the second is congruent to 5 modulo 6. We can use set builder notation and the roster method to specify the set \(A\) of all integers that are congruent to 3 modulo 6 as follows:

\begin{equation*} A = \left\{ a \in \Z \mid a \equiv 3 \pmod 6 \right\} = \{ \ldots, -15, -9, -3, 3, 9, 15, 21, \ldots \}\text{.} \end{equation*}

2.

Use the roster method to specify the set \(B\) of all integers that are congruent to 5 modulo 6.

\begin{equation*} B = \left\{ b \in \Z \mid b \equiv 5 \pmod 6 \right\} = \cdots \hspace{36pt}\text{.} \end{equation*}

Notice that \(15 \in A\) and \(11 \in B\) and that \(15 + 11 = 26\text{.}\) Also notice that \(26 \equiv 2 \pmod 6\) and that 2 is the smallest positive integer that is congruent to \(26 \pmod 6\text{.}\)

3.

Now choose at least four other pairs of integers \(a\) and \(b\) where \(a \in A\) and \(b \in B\text{.}\) For each pair, calculate \((a + b)\) and then determine the smallest positive integer \(r\) for which \(\left( a + b \right) \equiv r \pmod 6\text{.}\)

Note: The integer \(r\) will satisfy the inequalities \(0 \leq r \lt 6\text{.}\)

4.

Prove that for all integers \(a\) and \(b\text{,}\) if \(a \equiv 3 \pmod 6\) and \(b \equiv 5 \pmod 6\text{,}\) then \(\left( a + b \right) \equiv 2 \pmod 6\text{.}\)