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 a1 ∈ 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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 Spring '08
 Staff
 Natural number, Euclidean algorithm, Euclid, Fundamental theorem of arithmetic, Principal ideal domain

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