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MAT135_-_S11_-_Assignment_5_-_Solutions

# MAT135_-_S11_-_Assignment_5_-_Solutions - Assignment 5...

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Assignment 5 – Solutions 1. Prove that any two consecutive integers are coprime. Solution: Consider the consecutive integers n and n + 1. Because n ( - 1) + ( n + 1)(1) = 1, by Proposition 2.27(i), gcd( n, n + 1) = 1 and the 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? Solution: If a = b = 0 then the equation only has solutions if c = 0, and then any pair ( x, y ) Z will work. Otherwise suppose that the equation has a solution x 0 and y 0 . Then ax 2 0 + by 2 0 = c . Let d = gcd( a, b ) 6 = 0, so d | a and d | b . By Proposition 2.11(ii), d | ax 2 0 + by 2 0 , so d | c . The converse of this statement is not true, to see this let a = b = 1 and c = 3. Then gcd(1 , 1) = 1, which does divide 3. However, if x 2 + y 2 = 3, then x 2 3, so | x | ≤ 1 and | y | ≤ 1. None of these integer values give solutions. 3. (a) Prove that if n 3, then n ! + 3 is not prime. Solution: For n 3, n ! = n × ( n - 1) ×· · ·× 3 × 2 × 1, so 3 | n ! and n !

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MAT135_-_S11_-_Assignment_5_-_Solutions - Assignment 5...

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