homework-04

homework-04 - 1.2 # 2 3. 1.2 # 5 4. 1.2 # 7 (B) 1. 1.2 # 12...

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Modern Algebra and Geometry I Math 4000/6000 (92-321) Homework Assignments 4A and 4B – Due Friday, January 27 Homework is due at the start of class. Work that is not turned in at the beginning of class is considered late, and will not generally be accepted. Write your solutions in complete mathematical sentences. Always provide a complete justification for each solution, even if a problem does not explicitly ask you to do so. When using a result from the textbook as part of a solution, clearly identify the result you are using, e.g., “. . . so the statement is true for all n N by the Principle of Mathematical Induction,” or “. . . and 34 7 34 (mod 7) by Proposition 3.3 of Chapter 1.” This assignment consists of two parts that must be turned in separately. Problems in Part A will be marked by the course grader. Problems in Part B will be marked by the instructor. (A) 1. § 1.2 # 1b, 1c, 1d 2.
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Unformatted text preview: 1.2 # 2 3. 1.2 # 5 4. 1.2 # 7 (B) 1. 1.2 # 12 (Hint: If the rectangle has side lengths a and b , it might be useful to think about dierent ways you can write a in terms of b .) 2. 1.2 # 13. Note that in (a) it should be = | ab | , and in (b) it should be = | ab | /d . 3. Challenge problem: 1.2 # 20. Specically, when gcd( a,b ) = 1, determine an integer N such that it is possible to pay all postages greater than N cents, and then prove that it is possible to pay all postages greater than N cents. The integer N will be expressible via a formula in terms of a and b (e.g., one formula in terms of a and b is 3 ab + 5, though this isnt the correct formula for this problem). Additional practice problems: 1.2 # 6, 8, 9, 11, 14, 15, 21 These problems will not be collected, but cover material that you will be responsible for knowing. 1...
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