MODULAR ARITHMETIC

Worksheet 4

This is classwork, to be discussed in your groups. Each person should keep individual notes of the discussion.

1) Who says math is useless? You can become a credit card cheater! To begin, you need to know that some credit card numbers have different lengths and have different prefixes:

MasterCard is of length 16 and has a prefix of 51, 52, 53, 54, 55
VISA is of length 13 or 16 and has a prefix of 4
American Express is of length 15 and has a prefix of 34 or 37
Discover is of length 16 and has a prefix of 6011

All of the above credit cards use a mod 10 system to determine a check digit, and in all cases the check digit is the right-most digit in the number. Let's look at this alleged Mastercard number and test it for validity:

5368 2358 9683 1036

We look all the way at the right and see that the check digit is supposed to be a 6. We'll go through the procedure and see if we end up with a 6 ourselves.

Step 1: Start from the next to last digit (in this case a 3) and moving towards the left, multiply every other digit by 2.

3*2 = 6
1*2 = 2
8*2 = 16
9*2 = 18
5*2 = 10
2*2 = 4
6*2 = 12
5*2 = 10

Step 2: Sum the digits from Step 1.

6 + 2 + 1 + 6 + 1 + 8 + 1 + 0 + 4 + 1 + 2 + 1 + 0 = 33

Step 3: Sum all of the digits not originally multiplied by 2.

3 + 8 + 3 + 8 + 6 + 3 + 0 = 31
Step 4: Sum together the results from Step 2 and Step 3.
33 + 31 = 64

Step 5: Subtract the sum from the next highest multiple of 10.

70 - 64 = 6 (check digit)
Yay, we did it right! The check digit was supposed to be a 6, so the card is valid.

Okay, now see if you can make some changes in the card number given above and still get a valid credit card number. Test it carefully. You can change over to a different type of card, if you're feeling adventuresome.

2) Today is Monday. Your parents inform you that you are grounded for the next 1000 days, starting today. What day of the week will it be when you finally get back the keys to the minivan (i.e., you are finally ungrounded)?

3) Josh is writing a science fiction novel. He wants it to be as accurate as possible, so he goes to the trouble to figure out what day of the week it will be on September 8, 2044. How did he do that?

4) Mods may seem pretty weird, but actually you've been using the idea for years. Any time you refer to a number as "odd" or "even", you are using modular arithmetic. Explain the connection.

5) At swim practice, each swimmer does 200 laps. Fred starts with one lap of front crawl, then one lap of back crawl, and then one lap of sidestroke. (His knees hurt so he doesn't do breaststroke.) Then he restarts the cycle with front crawl again. Megan's cycle is: front crawl, back crawl, sidestroke and breaststroke. Taylor does: front crawl, back crawl, sidestroke, breastroke and butterfly. They all dive in at the same time and swim at exactly the same speed. (It's a fascist swim team, OK?)

a) What stroke will Fred do for his final lap?

b) How often (if ever) will all three swimmers be doing the same stroke at the same time?

c) Are there any combinations that are impossible? For example, maybe Megan and Fred never do sidestroke at the same time that Taylor does front crawl. Look for something like that.

d) Now introduce a new swimmer, Buffy, who does all five of the strokes that Taylor does along with a lap of dog paddle for strengthening her wrists. Answer questions b and c again.

6) (TRUE STORY) I am writing a computer program and I want the pen color to change each time the kid draws a new shape. I will be using colors from 2 (bright green) to 14 (yellow). I have to skip color 0 because it's black and color 1 because it's dark blue (can't be seen). Come up with a formula I can give the computer so that shape #1 will be color 2, shape #2 will be color 3, etc. up until I run out of colors. Then just restart with color 2, color 3 and so on in a cycle.
You may refer to the shape number as k, and give me a formula for color number in terms of k. Test it with a lot of examples to be sure it really works!

7) Make a Venn diagram to show the relationship between these classes:

4 mod 6
7 mod 9
7 mod 8
even numbers

It's pretty hard to draw. Make sure you test things from each category to be sure they really are possible. If you can't get the whole thing to work, draw the parts you are sure about and say what you know.

8) In previous classes, you probably studied graphs of quadratic functions written in vertex form:

Y = A(x - H)2 + K

You remember, perhaps, that each of the parameters A, H, and K controlled something different about the way the graph looked. Now take a similar approach to analyzing the graphs (or tables) of functions involving modular arithmetic. First, gather data by dividing the labor in your group and having different people graph different formulas involving mods. (You might want to stick with linear formulas to start with.) Then, come up with a way to write your mod formula in general form, and figure out what each parameter of the formula controls about the way the function behaves.

9) Expand on what you did for #7. Consider higher powers mixed with mods, and also more than one mod operation in a formula.

9) Generalize from what you did in #6. Figure out what types of relationships are possible between various classes. Then see if you can come up with a method to predict what that relationship will be simply by reading the class descriptions. That is, given class "a mod b" and class "c mod d", make a series of "if, then" statements to describe what the relationship between the two classes will be. Remember the three possibilities: intersection, subset, or disjoint.