Exam1Solutions - Math 302 Exam 1 Solutions Modern Algebra...

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Math 302 Modern Algebra Spring 2009 Exam 1 Solutions 1. (16 points) Prove Theorem 1.10 for positive integers: Every positive integer greater than 1 is the product of primes. This has symbolic form ( n > 1) [ n is a product of primes] . Note that a product of primes may include a single factor, so when we say “ n is a product of primes,” it includes the case that n is itself a prime. Proof: Use a proof by contradiction. Suppose the statement is false, so the negation is true. The negation is ( n > 1) [ n is not a product of primes] . Suppose then there exists a positive integer greater than 1 that is not the product of primes. We then need to obtain a contradiction. Let S be the set of all positive integers greater than 1 that are not the product of primes, so S = { n > 1 | n is not a product of primes } . Since we are supposing there exists a positive integer greater than 1 that is not the product of primes, then the set S is not empty. The Well Odering Axiom then implies S has a smallest element n 0 . Since n 0 S , then n 0 > 1 and n 0 is not the product of primes. Since n 0 is not the product of primes, then n 0 is not itself a prime. Therefore n 0 is composite. This implies n 0 = ab for some integers a and b , with 1 < a, b < m . Since both a and b are less than n 0 and greater than 1 and n 0 is the smallest element of S , then a / S and b / S . Therefore both a and b must be products of primes a = p 1 p 2 · · · p s and b = q 1 q 2 · · · q t , for some primes p 1 , p 2 , . . . , p s , q 1 , q 2 , . . . , q t . But then n 0 = ab implies n 0 = ( p 1 p 2 · · · p s )( q 1 q 2 · · · q t ) .
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