HW5.docx - HW 5 Exercise 6.5.1: Proving divisibility...

HW 5
Exercise 6.5.1: Proving divisibility results by induction (b).
b. Prove that for any positive integer n, 6 evenly divides 7
n
- 1.
Exercise 6.6.1: Proofs by strong induction - combining stamps (b). Note: You have to use
strong induction here. You will lose points if you use regular induction.
b. Prove that any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent
stamps.
Exercise 6.7.1: Proving the correctness of while loops using loop invariants (b).
b.
Exercise 7.1.2: Counting passwords made up of letters, digits, and special characters (a),
(b), (c).
Digits = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } = 10
Letters = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z } = 26
Special characters = { *, &, $, # } = 4 Total char = 40
a. string of length 6 = 40 6
=40 6
b. String of length 7 = 40
7
String of length 8 = 40 8
String of length 9 = 40
9
= 40 7 + 40 8 + 40 9
c. total char = 40
character = 26
= 14 with no characters
= 14(40 6 + 40 7 + 40 8 )
Exercise 7.1.4: Dividing up a print job (a), (b), (c).
100-page document
4 printers
a. 100 pages assigned to 4 printers = 4 100
b. first and last color pages, 2 printers print color/bw = 4 * 4
98
c. 25 assigned to same printer = 4 4
Exercise 7.2.2: Using the bijection rule to count palindromes (c).