math135-a5

math135-a5 - k integers could be.) 4. Either prove or give...

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Assignment 5 – Questions 1. Prove that any two consecutive integers are coprime. 2. Prove that the Diophantine equation ax 2 + by 2 = c does not have any integer solutions unless gcd( a,b ) | c . If gcd( a,b ) | c , does the equation always have an integer solution? 3. (a) Prove that if n 3, then n ! + 3 is not prime. (b) Prove that for every k P , k consecutive positive integers that are not prime can be found. (A good way to prove this is to explicitly show what these
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Unformatted text preview: k integers could be.) 4. Either prove or give a counterexample to, a 3 | b 3 if and only if a | b . 5. Find the prime factorization of gcd( a,b ) and of lcm( a,b ), where a = 25! and b = (5500) 3 (1001) 2 . 6. Find the number of positive integers whose prime factors are 2, 3 and 5 and which have exactly 20 positive divisors. 1...
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