b.inc(1) - B4 Suppose J ⊂ Z is an ideal Prove that there...

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MAS3300 Exam-B Prof. JLF King 4Jun2008 Note. This is an open brain, open ( pristine ) Sigmon- Notes exam. Please write each solution on a separate sheet of paper. Please be sure to write expressions unam- biguously e.g, the expression “1 /a + b ” should be bracketed either [1 /a ]+ b or 1 / [ a + b ]. Be careful with negative signs! B1: For free YMAssume that N is sealed under addition. Please prove that N is sealed under multi- plication. B2: Define a sequence b = ( b 0 ,b 1 ,b 2 ,... ) by b 0 := 0 and b 1 := 3 and b n +2 := 7 b n +1 - 10 b n , for n = 0 , 1 ,... . Use strong induction to prove, for all k > 0, that b k = 5 k - 2 k . (This is Exer2.15.2 in text.) B3: Prove Thm3.3 (P.15), the Division Alg. Thm.
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Unformatted text preview: B4: Suppose J ⊂ Z is an ideal. Prove that there exists a natnum b so that each member of J is a multiple of b . (I am asking you to prove a little less than Thm3.6d ( P.16 ). You’ll want to use the Div.Alg.Thm.) End of Exam-B B1: 60pts B2: 60pts B3: 60pts B4: 60pts Total: 240pts 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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