Homework:
Problems marked with an asterisk (*) are for extra credit.
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HW 1 (due 02/02 in P-143, Math Tower) [solutions]
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Section 1.1: 1, 2, 5, 6, 7
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Section 1.2: 1, 2, 3, 5, 6, 7, 8, 9*
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HW 2 (due 02/09 in P-143, Math Tower) [solutions]
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Section 1.3: Ex 1, 3, 4, 6, 7, 8
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Section 1.4: Ex 1, 2
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HW 3 (due 02/16 in P-143, Math Tower) [solutions]
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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
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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
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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.
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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
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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
- 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
x-y is an integer.
- Prove that M is an equivalence relation.
- Describe the equivalence classes of M.
- Section 4.1: Ex 1 (only the products π1π2, π2π3 and π2π1), 2, 3
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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
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HW 9 (due 04/6 in P-143, Math Tower) [solutions]
- Section 4:2: 12
- Section 4.3: 1
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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).
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HW 11 (due 04/22 in P-143, Math Tower) [solutions]
- Section 4.4: 7, 17*
- Section 5.1: 1, 3, 4, 5
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HW 12 (due 04/27 in P-143, Math Tower) [solutions]
- Section 5.1: 6, 10
- Section 5.2: 1, 2, 3, 5
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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