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hmwk_4

# hmwk_4 - (a a c â‰ b d(mod m(b ac â‰ bd(mod m Note If a = q...

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Discrete Structures Sept 12 2011 Assignment 4: Due at beginning of class Mon, Sept 26 Prof. Hopcroft *******Note: this homework is subject to the style guide on the website. Points will be deducted for homeworks not following the guidelines.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. (a) Use Euclid’s algorithm to compute the gcd of 495 and 210. Write out the steps. (b) What is the prime factorization of 495 and of 210? (c) Is your answer to part (a) correct? 2. Prove the following theorem Theorem: If a b (mod m ) amd c d (mod m ) then
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Unformatted text preview: (a) a + c â‰¡ b + d (mod m ) (b) ac â‰¡ bd (mod m ) Note: If a = q a m + r a and b = q b m + r b where r a < m and r b < m it is possible that r a + r b â‰¥ m . 3. Construct the multiplication table for arithmetic mod 7. 4. (Extended Euclidean Algorithm) What is multiplicative inverse of 400 mod 997? 5. (a) Prove for relatively prime a and b that if a divides bc , then a divides c . Hint : First show that there exist s and t such that sac + tbc = c and then argue that a divides c . (b) Give counter example when a and b are not relatively prime. 1...
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