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Unformatted text preview: Numbers and Polynomials Test 2 Solutions Answer FOUR questions. Be sure to give reasons for each step. 1. Prove that if n is a positive integer then 2 2 n- 1 +1 is a multiple of 3. What about 2 2 n + 1? What about 2 2 n- 1? 2. Let p be a prime number. Prove that if p divides the product ab of two integers a and b then p divides either a or b . 3. Let p be a prime number. Prove that there is no rational number t Q such that t 2 = p . 4. State the L east U pper B ound axiom. Prove that if r R then there exists n N such that n &gt; r . 5. Carefully define each of the following italicized expressions: (a) J is an ideal of Z ; (b) the integers a and b are coprime ; (c) p is a prime number; (d) the subset B R has a lower bound ; (e) R has the Archimedean property . Solutions : 1. Let n be the statement 2 2 n- 1 + 1 is a multiple of 3. Base step: 1 is true: indeed, 2 1 + 1 = 3....
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- Fall '08