MIT6_042JF10_assn03

MIT6_042JF10_assn03 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science September 21, 2010 Tom Leighton and Marten van Dijk Problem Set 3 Problem 1. [16 points] Warmup Exercises For the following parts, a correct numerical answer will only earn credit if accompanied by it’s derivation. Show your work. (a) [4 pts] Use the Pulverizer to ±nd integers s and t such that 135 s + 59 t = gcd(135 , 59). (b) [4 pts] Use the previous part to ±nd the inverse of 59 modulo 135 in the range { 1 ,..., 134 } . (c) [4 pts] Use Euler’s theorem to ±nd the inverse of 17 modulo 31 in the range { 1 ,..., 30 } . (d) [4 pts] Find the remainder of 34 82248 divided by 83. ( Hint: Euler’s theorem. ) Problem 2. [16 points] Prove the following statements, assuming all numbers are positive integers. (a) [4 pts] If a | b , then c , a | bc (b) [4 pts] If a | b and a | c , then a | sb + tc . (c) [4 pts] c , a | b ca | cb (d) [4 pts] gcd( ka,kb ) = k gcd( a,b ) Problem 3. [20 points] In this problem, we will investigate numbers which are squares modulo a prime number p . (a) [5 pts] An integer n is a square modulo p if there exists another integer x such that n x 2 (mod p ). Prove
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MIT6_042JF10_assn03 - 6.042/18.062J Mathematics for...

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