Homework:
Problems marked with an asterisk (*) are for extra credit.

HW 1 (due 02/02 in P143, Math Tower) [solutions]

Section 1.1: 1, 2, 5, 6, 7

Section 1.2: 1, 2, 3, 5, 6, 7, 8, 9^{*}

HW 2 (due 02/09 in P143, Math Tower) [solutions]

Section 1.3: Ex 1, 3, 4, 6, 7, 8

Section 1.4: Ex 1, 2

HW 3 (due 02/16 in P143, Math Tower) [solutions]

Section 1.4: Ex 3, 4, 5, 9
Section 1.5: Ex 1 (i, ii, iii, vi, vii), 2, 3
 Find all integral solutions of the equation:
 60x+18y=97
 21x+14y=147
 How many ways can change be made for one dollar, using each of
the following coins:
 dimes and quarters
 nickels, dimes and quarters

HW 4 (due 02/23 in P143, Math Tower) [solutions]
 Section 1.5: Ex 4, 5
 Section 1.6: Ex 1, 2, 3, 5, 6, 7, 8^{*}, 10

HW 5 (due 03/02 in P143, Math Tower) [solutions]
 Section 1.6: Ex 9^{*}, 12, 13
 Find the primes p and q if pq=4,386,607 and φ(pq)=4,382,136. Explain the method you have used.
 Are there any numbers n such that φ(n)=14? Explain!
 Find the remainder at division of 3^{1000} by 35.
 ^{*}Suppose that a cryptanalyst discovers a message P that is not relatively prime
to the enciphering modulus n=pq used in a RSA cipher. Show that the cryptanalyst
can actually factor n.

HW 6 (due 03/09 in P143, Math Tower) [solutions]
 ^{*} Compute φ(n) for the following values of n= 10!, 20! and 100!. Explain the method you have used.
 Section 2.1: Ex 1, 3, 4, 6
 Section 2.2: Ex 1, 2, 5, 9, 10
 Section 2.3: Ex 1

HW 7 (due 03/16 in P143, Math Tower) [solutions]
 Section 2.3: 2 (only c, e, d, f), 4, 6 (only the Hasse diagram), 7
 Let X= {1, 2, 3, 4, 5}. For each part, define a relation R on the set X so that R is
 reflexive and symmetric, but not transitive.
 symmetric and transitive, but not reflexive.
 reflexive, symmetric, transitive, and weakly antisymmetric.
 Let M be the relation on the real numbers R defined as follows:
for all x, y ∈R, xMy if and only if
xy is an integer.
 Prove that M is an equivalence relation.
 Describe the equivalence classes of M.
 Section 4.1: Ex 1 (only the products π_{1}π_{2}, π_{2}π_{3} and π_{2}π_{1}), 2, 3

HW 8 (due 03/30 in P143, Math Tower) [solutions]
 Section 4.1: 4, 5, 6
 Write the permutations π_{1}, π_{2} and π_{4} in the exercise 1 of section 4.1 as a product of disjoint cycles.
 Section 4.2: 1, 2, 3, 5, 6, 7

HW 9 (due 04/6 in P143, Math Tower) [solutions]
 Section 4:2: 12
 Section 4.3: 1

HW 10 (due 04/13 in P143, Math Tower) [solutions]
 Section 4.3: 2, 3, 5, 6, 8
 Describe the group of symmetries of a regular hexagon. How many symmetries does it have? What do they look like?
What relationships are there among them?
 Section 4.4: 1, 3 ((i), (iii) and (v) only).

HW 11 (due 04/22 in P143, Math Tower) [solutions]
 Section 4.4: 7, 17^{*}
 Section 5.1: 1, 3, 4, 5

HW 12 (due 04/27 in P143, Math Tower) [solutions]
 Section 5.1: 6, 10
 Section 5.2: 1, 2, 3, 5

HW 13 (due 05/4 in P143, Math Tower) [solutions]
 Section 5.3: 1, 2, 4, 7, 8, 9
 Section 5.4: 2, 4, 5