I briefly review the mod 3 addition and multiplication tables. I then asked students to construct addition and multiplication tables for mod 7 arithmetic. This took a little longer than I expected but students were asking good questions and comparing their work. As they got going on the work I could see that the tables were really starting to make sense to them.
I projected a couple of student work examples on the board and students asked questions about discrepancies or points of confusion. I pointed out the cyclical nature of the addition tables and I pointed out that in the multiplication tables, for every non-zero column or row, each value from 0-6 only appeared once. I also reviewed that the arithmetic we were seeing possessed all the same properties as working with real numbers. Properties such as the associative property, the commutative property, and the distributive property all worked.
We had been looking at arithmetic modulo a prime number for the reason that this arithmetic forms an algebraic field. All the properties for operations with real numbers also are present with these finite algebraic fields. The next task was to look at what happens when the modulus is a composite number rather than a prime number.
I asked students to construct addition and multiplication tables for a composite number. The slide asked for mod 6 tables but I told the students they could use any composite number. Some created tables for mod 4 and others chose mod 6. At this point students were much more comfortable constructing the tables and they completed the task in a more timely manner.
We looked at sample tables for mod 4 and mod 6. It was readily apparent that the addition tables behaved as before, showing a cyclical nature. I could also point out that one set of diagonals were also cyclical. Everything looked good from the addition perspective.
The multiplication tables posed a different issue. Students now saw repeated patterns in some of the rows and columns. I asked students to consider why this would happen. One student thought it might have to do with the factors of the number. When pressed further, the suggestion was that values like 2,3, and 4 in the mod 6 table had a common factor with 6. This is precisely what is happening.
This allowed me to revisit the cipher function of C(x) = 2x + 5 (mod 26). I asked students if they remembered using this cipher function and finding that different letters were mapped to the same value. Many shook their heads in agreement. The reason this happened is that the cipher function's slope of 2 and 26 have a common factor. So, if we have a cipher function C(x) = Ax + B (mod m), we will not get unique mappings unless gcd(A, m) = 1.
I then revisited Congruence Property 1:
If a ≡ b (mod m) then ak ≡ bk (mod m) for any integer k.
This is a property that we proved in class. I used this as the basis to argue that if 3 x 2 ≡ 2 then 3 ≡ 1. As I explained in an earlier post, I cheated on this piece as I wanted students to focus on the concept and not get bogged down in details. Today I was able to formally discuss the converse of this congruence property, specifically,
If ka ≡ kb (mod m) then a ≡ b (mod m) for any integer k, provided gcd(k, m) = 1.
The requirement for gcd(k, m) = 1 comes from the need that m|k(a-b). If k and m are not relatively prime, we cannot guarantee that m|(a-b).
Class closed with students summarizing their thoughts about modular arithmetic in their notes.
Visit the class summary for a student's perspective and to view the lesson slides.
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