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final-fall94

# final-fall94 - 7 x 121 y = 1 Hence ﬁnd the multiplicative...

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Name: S.S.N: MAS 3300 – FINAL EXAM Fall 1994 In giving proofs make sure your reasoning is clear. The questions have equal value. Write on ONE side of your paper. 1. PROVE If a,b R then a ( - b ) = - ( ab ). 2. Use mathematical induction to prove that 1 1 · 2 + 1 2 · 3 + ··· + 1 n ( n + 1) = 1 - 1 n + 1 for all integers n 1. 3. PROVE If a,b,c Z , a | b and b | c then a | c . 4. PROVE that there are inﬁnitely many primes. 5. Use uniqueness of prime factorizations to prove that 18 is irrational. 6. PROVE that if Z n is a ﬁeld then n must be prime. 7. Find integers x and y such that
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Unformatted text preview: 7 x + 121 y = 1 . Hence ﬁnd the multiplicative inverse of 7 in Z 121 . 8. PROVE If z 1 , z 2 ∈ C then z 1 z 2 = z 1 z 2 . 9. Use the Rational Root Theorem 7.14 to prove that 3 √ 2 is irrational. 10. PROVE Q [ √ 2] ∩ Q [ √ 3] = Q . 11. PROVE (i) If r,s ∈ R and 0 ≤ r < 1 and 0 ≤ s < 1 then r + s < 1 + rs . (ii) If z 1 , z 2 ∈ C , | z 1 | < 1 and | z 2 | < 1 then ± ± ± ± z 1-z 2 1-z 1 z 2 ± ± ± ± < 1 . 1...
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