Answers

Counting On A Clock

1) 15 (mod 12) = 3

2) 29 (mod 12) = 5

3) 7 (mod 12) = 7

4) 36 (mod 12) = 0

5) -4 (mod 12) = 8

6) 10 (mod 4) = 2

7) 12 (mod 7) = 5

8) 35 (mod 12) = 11

9) 2 (mod 25) = 2

10) 8 (mod 4) = 0

11) -5 (mod 10) = 5

12) -13 (mod 6) = 5

Challenge Problems
a) 5, 8, 11, etc. (There are other correct answers.)
b) The remainder of 4 and 7 when divided by 3 is the same, 1. (Answers may vary on this question.) Positive numbers: 10, 13, 16, etc. Negative numbers: -2, -5, -8, etc. (There are other correct answers for both positive and negative numbers.)


Modular Arithmatic

1) 2 + 7 (mod 4) = 1

2) 3 + 8 (mod 7) = 4

3) 1 + 1 (mod 9) = 2

4) 2 + 3 (mod 5) = 0

5) 4 + 0 (mod 6) = 4

6) 7 - 2 (mod 2) = 1

7) 8 - 4 (mod 3) = 1

8) 4

9) 1

10) 3

Challenge Problems
2 * 3 (mod 4) = 2
4 * 2 (mod 7) = 1
3 * 4 (mod 5) = 2


Modular Multiplication

1) 3 * 2 (mod 4) = 2

2) 7 * 3 (mod 5) = 1

3) 2 * 2 (mod 4) = 0

4) 10 * 3 (mod 7) = 2

5) 0

6) 0

7) 0

Challenge Problems
a) 2
b) 2
c) Multiplicative inverse of 3 in mod 7 is 5. Multiplicative inverse of 3 in mod 5 is 2. Not the same. This is because they are in different mods or they are dealing with different divisors or different cycle lengths (answers may vary).


Multiple Operations

1) 5 * 3 + 2 (mod 6) = 5

2) 2 * 1 + 4 (mod 3) = 0

3) 3 * 4 - 5 (mod 2) = 1

4) 5 * 0 + 2 (mod 8) = 2

5) 5 * 0 - 2 (mod 8) = 6


Linear Equations

1) 2

2) 4

3) 3

4) 8

5) 1

6) 2

7) 2

8) 1


Graphing

Y = 2x + 3 (mod 5)
X Y
-5 3
-4 0
-3 2
-2 4
-1 1
0 3
1 0
2 2
3 4
4 1
5 3


Students' plots should match the plot that the model makes.

The graph repeats because it is in mod or can't go above a certain number (answers may vary).

Predictions
X Y
-10 3
-9 0
-8 2
-7 4
-6 1
6 0
7 2
8 4
9 1
10 3


Predictions should be repeats of the original graph on each side of the plot.

Y = 3x + 2 (mod 4)
X Y
-4 2
-3 1
-2 0
-1 3
0 2
1 1
2 0
3 3
4 2


Predictions
X Y
-8 2
-7 1
-6 0
-5 3
-4 2
4 2
5 1
6 0
7 3
8 2



Solving Systems of Equations

Number 1
X Y
-5 2
-4 3
-3 0
-2 1
-1 2
0 3
1 0
2 1
3 2
4 3
5 0


Number 2
X Y
-5 0
-4 1
-3 0
-2 1
-1 0
0 1
1 0
2 1
3 0
4 1
5 0


Plots should match the model.

3) Solutions are X = -3, -2, 1, 2, 5

4) Yes there is a pattern. Two solutions next to each other and then a gap of two before the next solution (count three and that one is a solution). X = 6, 9, 10, 13, 14, etc (There are more correct solutions.).

Number 5
X Y
-5 2
-4 0
-3 1
-2 2
-1 0
0 1
1 2
2 0
3 1
4 2
5 0


Solutions are X = -5 and X = 5.

There are not enough points to find a pattern and predict other solutions.

^M