1. Complete the addition and multiplication tables for arithmetic modulo 5:

 

+ 5	0	1	2	3	4
0	0	1	2	3	4
1	1			4	0
2	2
3
4


 

· 5	0	1	2	3	4
0	0	0	0	0	0
1	0	1	2	3	4
2	0			1
3
4


 
2. Exercise 12.1.5 from the notes. See the worksheet below.

 
3. Extra credit problems  (Click here or scroll down).

 

  Worksheet for Exercise 12.1.5

(Computations are to be done modulo 5.)
 

Line	Equation	Points	Corresponding to:
Line 1	x = 0	(0,0) (0,1) (0,2) (0,3) (0,4)	A B C D E
Line 2	x = 1	(1,0) (1,1) (1,2) (1,3) (1,4)	F G H I J
Line 3	x = 2		K L M N O
Line 4			P Q R S T
Line 5			U V W X Y
Line 6	y= 0	(0,0) (1,0) (2,0) (3,0) (4,0)	A F K P U
Line 7	y= 1	(0,1) (1,1) (2,1) (3,1) (4,1)	B G _ _ _
Line 8			_ _ _ _ _
Line 9			_ _ _ _ _
Line 10			_ _ _ _ _
Line 11	y= x	(0,0) (1,1) (2,2) (3,3) (4,4)	A G M S Y
Line 12	y = x + 1	(0,1) (1,2) (2,3) (3,4) (4,0)	B H N T F
Line 13	y= x + 2	(0,2) (1,3) (2,__) (3,__) (4,__)	C I _ _ _
Line 14		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 15		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 16	y = 2x	(0,0) (1,2) (2,4) (3,1) (4,3)	A H O Q _
Line 17	y = 2x + 1	(0,1) (1,3) (2,0) (3,__) (4,__)	B I _ _ _
Line 18		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 19		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 22		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 21	y = -x	(0,0) (1,4) (2,3) (3,__) (4,__)	_ _ _ _ _
Line 22	y = -x + 1	(0,1) (1,0) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 23	y = -x + 2	(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 24		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 25		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 26	y = -2x	(0,0) (1,3) (2,1) (3,__) (4,__)	_ _ _ _ _
Line 27	y = -2x+1	(0,1) (1,4) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 28		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 29		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _
Line 30		(0,__) (1,__) (2,__) (3,__) (4,__)	_ _ _ _ _

 

Extra Credit Problems

While the credit scales are not yet completely determined, it is clear that problem 2 will count for about three times as many points as problem 1.

1. Fill in the remaining lines in the diagram of mod 3 plane geometry shown below. The lines y= x and y= x + 2 are the ones that are shown in color. (To make a more symmetric diagram, note that the lines y = 2x + c, nominally given as having slope = 2, can also be considered as having slope = -1. Indeed, since 3 = 0, we can consider 2 = -1.)
   Note: The lines in mod 3 plane geometry are finite sets of points, so the lines in the diagram are just an abstract depiction that shows which sets constitute lines in this geometry.



 

2. Make a similar diagram for mod 5 plane geometry.

(Suggestion: You may need to use several colors to make lines with various slopes show more clearly.)