test3-fall94

# test3-fall94 - 216 using the prime factorizations 2 Let...

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Name: S.S.N: MAS 3300 – Test 3 Fall 1994 When giving proofs quote the relevant axioms and results. The questions have equal value. Do SEVEN questions. If you do more than seven ques- tions please indicate which seven you want graded as test questions. Any other question attempted will be marked as a bonus question. You may use your unmarked printed course notes. Calculators are allowed. Write on ONE side of your paper. 1. (i) Use the Euclidean algorithm to ﬁnd the gcd(600 , 216). (ii) Find the prime factorizations of 600 and 216. (iii) Find the gcd(600
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Unformatted text preview: , 216) using the prime factorizations. 2. Let a,b > 1 be integers and suppose p is prime. Prove that if p | ( ab ) then either p | a or p | b . 3. Prove that there are inﬁnitely many primes. 4. Prove that if a ∈ Q and a 6 = 0 then a-1 ∈ Q . 5. Prove that √ 3 √ 5 is irrational. 6. Prove that if a,b > 1 are integers and a 5 | b 4 then a | b . 7. Prove that 11 + 1 √ √ 3+ √ 7 is irrational. 8. Prove that log 2 10 is irrational. 9. Let a,b > 1 be integers. Prove that gcd( a,b )lcm( a,b ) = ab. 1...
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