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110ahw2 - Math 110A Homework#2 David Wihr Taylor Summer...

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Math 110A Homework #2 David Wihr Taylor Summer 2006 1. Problem 1.3.1: Express each number as the product of primes: (a)5040, (b) - 2345, (c) 45670, (d) 2042040. Answer: To my knowledge there is no fast algorithm to factor primes that doesn’t use quantum computing (see Shore’s algorithm if you want to see how to factor on a quantum computer). If there were some conventional algorithm, then all computer encryption would be unsafe (same thing if anyone ever builds a quantum computer)! However, there is an algorithm that runs as a polynomial function of the size of the number you input that tells you if your number is prime. All this basically doesn’t help you do this problem, but it’s interesting stuff anyway. Mathematica tells me: (a) 5040 = 2 4 · 3 2 · 5 · 7 (b) - 2345 = - 1 · 5 · 7 · 67 (c) 45670 = 2 · 5 · 4567 (d) 2042040 = 2 2 · 3 · 5 · 11 · 13 · 17. 2. Problem 1.3.9: Claim. If p is prime and ( a, b ) = p then ( a 2 , b 2 ) = p 2 Proof. First we notice that if p divides both a and b then p 2 divides both a 2 and b 2 . Now let’s express a and b by their prime factorizations a = p r 1 1 · · · p r k k and b = q s 1 1 · · · q s l l .
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