test2-summer01

# test2-summer01 - (iii Find gcd(819 165 using the Euclidean...

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Name: CODEWORD: MAS 3300 – Test 2 Summer A 2001 Give proofs of the following quoting the relevant axioms and results. Proofs should be written in a proper and coherent manner. Write so that anyone in the class can follow your work. The questions have equal value. Write on ONE side of your paper. 1. Suppose a , b , c Z . If a | b and a | c then a | ( b + c ). 2. Suppose a , b Z . Then J a,b = { ka + ‘b : k, ‘ Z } is an ideal of Z . 3. Suppose a , b , c Z . If a | ( bc ), and a and b are relatively prime, then a | c . 4. (i) Find prime factorizations of 819 and 165. (ii) Find gcd(819 , 165) using prime factorizations.
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Unformatted text preview: (iii) Find gcd(819 , 165) using the Euclidean algorithm. 5. Find integers x , y such that 819 x + 165 y = 3 . 6. There are inﬁnitely many primes. 7. 4 √ 8 is irrational. 8. For every real number b , there is a natural number n ∈ N such that b < n . HINT: Assume by way of contradiction that the result is false. Then apply LUB to the set N . 9. Suppose a , b ∈ N and a,b ≥ 1. Then gcd( a,b )lcm( a,b ) = ab. HINT: Use Theorem 3.27. 1...
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