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Beginning Activity Beginning Activity 2: Some Work with Congruence Modulo n
A4US

1.

Let n be a natural number and let a and b be integers.

(a)

Write the definition of “a is congruent to b modulo n,” which is written a≡b(modn).

(b)

Use the definition of “divides” to complete the following:

When we write a≡b(modn), we may conclude that there exists an integer k such that ….

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:

A={a∈Z∣a≡3(mod6)}={…,−15,−9,−3,3,9,15,21,…}.

2.

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

B={b∈Z∣b≡5(mod6)}=⋯.

Notice that 15∈A and 11∈B and that 15+11=26. Also notice that 26≡2(mod6) and that 2 is the smallest positive integer that is congruent to 26(mod6).

3.

Now choose at least four other pairs of integers a and b where a∈A and b∈B. For each pair, calculate (a+b) and then determine the smallest positive integer r for which (a+b)≡r(mod6).

Note: The integer r will satisfy the inequalities 0≤r<6.

4.

Prove that for all integers a and b, if a≡3(mod6) and b≡5(mod6), then (a+b)≡2(mod6).