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hw4 - mod 11 7 For each nonzero class x =[1[2[10 write out...

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MAT 312/AMS 351 – Fall 2010 Homework 4 1. p. 43 Exercise 1. 2. p. 43 Exercise 3. Consider also the numbers 9 and 11. 3. Given positive integers a and b , suppose there exist integers k and such that 1 = ka + ℓb . Show that a and b must be relatively prime. 4. Suppose a positive number n > 1 is relatively prime to every number strictly between 1 and n . Show that n must be a prime. 5. Using the previous 2 exercises, or otherwise, prove that in the number system of integers mod n , if every class has a multiplicative inverse then n must be a prime. 6. Write out the multiplication table for the set of congruence classes
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Unformatted text preview: mod 11. 7. For each nonzero class x = [1] , [2] , . . . , [10], write out the sequence of powers of x mod 11. For example [5] 1 = [5] , [5] 2 = [25] = [3] , [5] 3 = [3 · 5] = [15] = [4], etc. Stop when you get to [1]. 8. The number of distinct powers of x (counting x = [1]) is called the (multiplicative) order of x . What are the di±erent orders occurring in your list? What numerical property do these orders share? 1...
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