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test2-fall92

# test2-fall92 - = n n 1 2 5 Show that J a:= ka a ∈ Z is an...

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Name: S.S.N: MAS 3300 – Test 2 Fall 1992 Give proofs of the following quoting 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. 1 | a for all a Z . 2. Let J be a nonempty ideal of Z . Then 0 J . 3. If a | b and a | c then a | ( b + c ). 4. Use mathematical induction to show that for any positive integer n 1 + 2 + ··· + n
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Unformatted text preview: = n ( n + 1) 2 . 5. Show that J a := { ka : a ∈ Z } is an ideal of Z . 6. Use the Well Ordering Principle to show that there is no integer α such that < α < 1 . 7. If a | bc and a and b are relatively prime then a | c . 8. Let a , b be both not zero. Let d = gcd( a,b ). There are integers x , y such that ax + by = c if and only if d | c . 1...
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