Appendix D Hints and Answers to Selected Exercises
1 Preliminaries
1.4 Exercises
More Exercises
1.4.18.
1.4.20.
1.4.22.
1.4.24.
1.4.28.
2 The Integers
2.4 Exercises
2.4.1.
2.4.3.
2.4.9.
Hint.
Follow the proof in Example 2.1.4.
2.4.11.
2.4.17. Fibonacci Numbers.
Hint.
For Item 2.4.17.a and Item 2.4.17.b use mathematical induction. Item 2.4.17.c Show that and Item 2.4.17.d Use part Item 2.4.17.c. Item 2.4.17.e Use part Item 2.4.17.b and Exercise 2.4.16.
2.4.19.
2.4.23.
Hint.
Let Then since By the Principle of Well-Ordering, contains a least element To show uniqueness, suppose that and for some By the division algorithm, there exist unique integers and such that where Since and divide both and it must be the case that and both divide Thus, by the minimality of Therefore,