hw10 - nitely many primes, say, p 1 ,p 2 ,...,p n . Form...

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MATH 210-02 (Fall 2007) Dr. Kwong Homework 10 (60 Points) Due Tuesday, 12/4/2007, before noon . No late homework will be accepted. Instructions : Be sure to explain how you obtain your answers, and write up your solutions neatly and clearly, in a manner that could be understood by your fellow classmates. Sloppy work will be returned ungraded. Be sure to start with a draft before copying the final version to the sheets to be handed in. 1. [8 points] Use (Extended) Euclidean Algorithm to find gcd(4485 , 9802), and find the linear combination that gives this gcd. 2. [6 points] Let m and n be two integers that are relatively prime. What are the possible values of gcd(4 m + 7 n, 7 m - 4 n )? 3. [6 points] Here is the outline of another proof that there are infinitely many primes. Rewrite the proof in your own words: The idea is almost identical to the one we studied in class. Suppose there are only
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Unformatted text preview: nitely many primes, say, p 1 ,p 2 ,...,p n . Form the number N = p 2 p 3 p 4 p n + p 1 p 3 p 4 p n + p 1 p 2 p 4 p n + + p 1 p 2 p 3 p n-1 . Note that N is the sum of n numbers, and the i th term is the product of all the primes except p i . Show that N cannot be composite, hence N must be prime. Apply the same idea we used in class to deduce a contradiction. 4. [8 points] Let p be a prime number. Show that p is irrational. Be sure to explain each step of your argument. 5. [6 points] Find the sum and product of 207 and 114 in Z 23 . 6. [8 points] Use repeated squaring to compute 4 41 (mod 11). 7. [6 points] Compute 23-1 (mod 37). 8. [6 points] Solve the equation 23 x + 12 = 3 over Z 37 . 9. [6 points] The integer a has the property a 6 1 (mod 13). Find ( a 2 )-1 (mod 13). Explain!...
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This homework help was uploaded on 04/19/2008 for the course MATH 210 taught by Professor Kwong during the Fall '08 term at SUNY Fredonia.

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