Assignment III Math for Poets 2006

Question 1: Do problems 33,34 on page 93 of the book (= problems 8,9 on page 94 of the first edition).
For computations modulo 5 we can make an addition and multiplication table as follows:
Addition:

		0	1	2	3	4
0		0	1	2	3	4
1		1	2	3	4	0
2		2	3	4	0	1
3		3	4	0	1	2
4		4	0	1	2	3

Multiplication:

		0	1	2	3	4
0		0	0	0	0	0
1		0	1	2	3	4
2		0	2	4	1	3
3		0	3	1	4	2
4		0	4	3	2	1

Question 2: Looking at the tables can you find a number which, added to 3, is 2 modulo 5?
Question 3: Looking at the tables can you find a number which, multiplied with 3, is 2 modulo 5?
Question 4: We now look at numbers modulo 12. Can you find a number which, multiplied by 7 gives 1 modulo 12?
You can either use a multiplication table or, more simply, trial and error.
Question 4: A flea jumps back and forth on the numerals 1,2,3...,12 of a clock. It can only take jumps of length 7.
For example, it can jump from 1 to 8, from 4 to 11 or, jumping backward, from 4 to 9.
What is the minimal number of jumps that the flea requires to go from 1 to 2? Describe the direction in which it should move.
Describe your solutions in such a clear way that you colleague students are able to understand them from your written text.