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# c.inc - AT AF Nei q x is irrational AT AF Nei Essay...

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Numbers & Polys MAS3300 Exam-C Prof. JLF King 4Jun2008 Note. This is an open brain, open ( pristine ) Sigmon- Notes exam. Please write solutions for C3 , C4 , C5 on separate sheets of paper. Write expressions unambigu- ously e.g, “1 /a + b ” should be bracketed either [1 /a ] + b or 1 / [ a + b ]. Be careful with negative signs! Every if ” must be matched by a “ then .” Short answer; show NO work. C1: a Let Γ := Gcd(165 , 63) and find particular integers so that 165 A + 63 B = Γ. Then Γ= , A = , B = . b Note that Gcd(15 , 21 , 35) = 1. Find particular integers S, T, U so that 15 S + 21 T + 35 U = 1: S = , T = , U = . [ Hint: Gcd ( Gcd(15 , 21) , 35 ) = 1. ] C2: Let AT AF Nei mean “Always True”, “Always False”, and “Neither”, respectively. Let q represent a general non-zero rational number and let x, y denote general irrationals. Please circle the best answer. The product x · y is irrational. AT AF Nei The sum x + y is irrational. AT AF Nei q · x is irrational. AT AF Nei q + x is irrational. AT AF Nei Essay questions.
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Unformatted text preview: AT AF Nei q + x is irrational. AT AF Nei Essay questions. Please write in complete sentences. C3: Prove that √ 15 is irrational. Now prove that √ 3 + √ 5 is irrational. C4: Consider two posints A and B . Prove directly, using WO, that there is a posint D st. D Z = A Z + B Z . [ Hint: Don’t forget to ﬁrst show that A Z + B Z has a non-zero element.] C5: Prove that there are inﬁnitely many prime num-bers. End of Exam-C C1: 70pts C2: 40pts C3: 80pts C4: 80pts C5: 80pts Total: 350pts Print name ............................................. Ord: Honor Code: “I have neither requested nor received help on this exam other than from my professor (or his colleague) .” Signature: ............................................
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