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hmwk_4_sol

# hmwk_4_sol - Discrete Structures Assignment 4 Solutions Due...

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Discrete Structures Sept 12 2011 Assignment 4 Solutions: 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. gcd (495 , 210) = gcd (495 - 210 , 210) = gcd (285 , 210) = gcd (285 - 210 , 210) = gcd (75 , 210) = gcd (75 , 210 - 75) = gcd (75 , 135) = gcd (75 , 135 - 75) = gcd (75 , 60) = gcd (75 - 60 , 60) = gcd (15 , 60) = 15 (b) What is the prime factorization of 495 and of 210? 495 = 3 * 3 * 5 * 11 210 = 2 * 3 * 5 * 7 (c) Is your answer to part (a) correct? Yes. 2. Prove the following theorem Theorem: If a b (mod m ) amd c d (mod m ) then (a) a + c b + d (mod m ) Proof. If a b (mod m ), then a = b + im , for some integer i . Similarly, c = d + jm . Now: a + c = b + im + d + jm = b + d + ( i + j ) m b + d + ( i + j ) m (mod m ) b + d (mod m ) (b) ac bd ( (mod m )) Proof. If a b (mod m ), then a = b + im , for some integer i . Similarly, c = d + jm . Now: ac = ( b + im )( d + jm ) = bd + dim + bjm + ijm 2 bd + dim + bjm + ijm 2 (mod m ) bd (mod m ) 1

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hmwk_4_sol - Discrete Structures Assignment 4 Solutions Due...

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