# MAT 312/AMS 351 Applied Algebra

### Homework:

Problems marked with an asterisk (*) are for extra credit.
• HW 1 (due 02/02 in P-143, 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 P-143, 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 P-143, 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:
1. 60x+18y=97
2. 21x+14y=147
• How many ways can change be made for one dollar, using each of the following coins:
1. dimes and quarters
2. nickels, dimes and quarters
• HW 4 (due 02/23 in P-143, 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 P-143, 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 31000 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 P-143, 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 P-143, 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
1. reflexive and symmetric, but not transitive.
2. symmetric and transitive, but not reflexive.
3. reflexive, symmetric, transitive, and weakly antisymmetric.
• Let M be the relation on the real numbers R defined as follows: for all x, yR, xMy if and only if x-y is an integer.
1. Prove that M is an equivalence relation.
2. Describe the equivalence classes of M.
• Section 4.1: Ex 1 (only the products π1π2, π2π3 and π2π1), 2, 3
• HW 8 (due 03/30 in P-143, Math Tower) [solutions]
• Section 4.1: 4, 5, 6
• Write the permutations π1, π2 and π4 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 P-143, Math Tower) [solutions]
• Section 4:2: 12
• Section 4.3: 1
• HW 10 (due 04/13 in P-143, 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 P-143, Math Tower) [solutions]
• Section 4.4: 7, 17*
• Section 5.1: 1, 3, 4, 5
• HW 12 (due 04/27 in P-143, Math Tower) [solutions]
• Section 5.1: 6, 10
• Section 5.2: 1, 2, 3, 5
• HW 13 (due 05/4 in P-143, Math Tower) [solutions]
• Section 5.3: 1, 2, 4, 7, 8, 9
• Section 5.4: 2, 4, 5