Thursday 22 September class

Topics:
• Activity: computing 96³ mod 7  two ways:
  By first computing 96³ and then reducing mod 7
  By first reducing 96 mod 7, then raising that number to the 3rd power (and then reducing again)
This showed by example that we can reduce the base of a whole number exponent before exponentiation in modular arithmetic. (This is not a proof, but an example.)

While showing the solution on the board, I showed how you can use a calculator to find a remainder/residue. The details are in this post.

We contrast this with the computation of 7¹⁹⁴² mod 100 that we did last time. The difference was that there we would want to reduce the exponent, but reducing mod 100 does not work. (I could not show this directly because the numbers involved are too big, but I made an example of 3¹⁵ mod 5 later.)

This led to a discussion of a heuristic that connects to the thinking involved in the way we found the value of 7¹⁹⁴² mod 100. I have posted those notes separately.

Next  was to construct some addition and multiplication tables for modular arithmetic. Since the modular number systems are finite, we can construct complete tables for them. Fixing the modulus m,  we show all the residues for that modulus along the top and side of the table: then the entry at each intersection of a row and column is the sum or product of those two residues, reduced mod m. We made the following addition and multiplication tables:

Mod 2: Addition

+	0	1
0	0	1
1	1	0

Mod 2: Multiplication

•	0	1
0	0	0
1	0	1

Mod 3: Addition

+	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

The tables for mod 2 allow us to prove the results for the questions in last time's warmup, and others, since an integer is even if (and only if) it is congruent to 0 mod 2, and it is odd if (and only if) it is congruent to 1 mod 2. Therefore, for example, we can translate:

0+0 ≡ 0 (mod 2) means that "even plus even = even"

1+1 ≡ 0 (mod 2) means that "odd plus odd = even"

But we get even more: 0+1 ≡ 1 (mod 2) means that "even plus odd = odd"

Go through the tables and see what else they prove!


Homework:
• Journal assignment: Write about the group activity (computing 96³ (mod 7) two ways): what your group did, and what you learned about modular arithmetic from this.
• Problems to work:
1)  Construct a multiplication table for multiplication mod 3

2) Construct an addition table and a multiplication table for addition and multiplication mod 5

3) The moduli we are most interested in are prime numbers. To see one reason for this, construct the multiplication table for multiplication mod 4. (It may not be obvious from the table that there is a problem: you might want to think about the zero product property from algebra. More next time.)

4) Translate the 8 entries in the addition and multiplication tables mod 2 into statements about the sums and products of even and odd numbers: three of them are translated in the notes above.