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test3-fall92 - n such that na> b Proof Suppose not ie na...

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Name: S.S.N: MAS 3300 – Test 3 Fall 1992 In giving proofs quote the relevant axioms and results. The questions have equal value. Write on ONE side of your paper. Throughout this test roman letters a , b , . . . refer to integers. 1. (i) Find prime factorizations of 221 and 85. (ii) Find gcd (221 , 85) using prime factorizations. (iii) Find gcd (221 , 85) using the Euclidean algorithm. 2. (i) Find integers x and y such that 221 x + 85 y = 17 . Hint. Use your work done in 1(iii). (ii) Find integers x and y such that 221 x + 85 y = 170 . 3. Prove Q is closed under addition. 4. Prove or disprove. If a and b are irrational then a + b is irrational. 5. Prove 20 is irrational. You may NOT assume Theorem 4.3(c). 6. Prove there are infinitely many primes. 7. Complete the following proof. 4.7 Archimedean Property . For each a, b R with a > 0 there exists and integer
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Unformatted text preview: n such that na > b. Proof. Suppose not. ie. na ≤ b for all integers n . Then since a > 0 we have n ≤ b a for all n ∈ Z . Hence b a is an upper bound for Z . By LUB Z must have a least upper bound u . ie. for some u ∈ R n ≤ u for all n ∈ Z . 1 2 8. Prove √ 2 + √ 3 + √ 5 is irrational. HINTS: (i) Suppose not; i.e. suppose √ 2 + √ 3 + √ 5 is rational. (ii) By squaring show that this implies √ 6 + √ 10 + √ 15 is rational. (iii) By squaring again show that this implies 5 √ 6 + 3 √ 10 + 2 √ 15 is rational. (iv) Hence show that this implies that 3 √ 15 + 2 √ 10 is rational. (v) Hence show that this implies √ 150 is rational to obtain a contradiction....
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