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Fall2006 M1

# Fall2006 M1 - 70 266 2016 = − v u with the additional...

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1. Let A be the statement “If n is an odd integer, then n 2 + n is an even integer. (a) (2 marks) Is A TRUE or FALSE? Justify your answer. (b) (3 marks) State the converse of A . Is the converse of A TRUE or FALSE? Justify your answer. (c) State the contrapositive of A . 2. (a) (2 marks) Let Z b a , . Give the definition of the statement “ a divides b ”. (b) (4 marks) Let Z y x c b a , , , , . Prove that is a | b and a | c , then cy bx a + | . (c) (3 marks) Consider the statement “For all Z c b a , , , if cy bx a + | for all integers x and y , then a | b and a | c .” Is this statement TRUE or FALSE? Prove your answer. (d) (2 marks) Prove that the statement “For all Z c b a , , , if there exist integers x and y such that cy bx a + | , then a | b and a | c .” is FALSE by finding a counterexample. 3. (a) (8 marks) Determine the complete solution to the linear Diophantine equation 70 266 2016 = + y x (b) (2 marks) Write down the complete solution to the linear Diophantine equation 70 266 2016 = v u (c) Determine all non-negative solutions to the linear Diophantine equation

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Unformatted text preview: 70 266 2016 = − v u with the additional property that 360 ≤ + v u . 4. (9 marks) A sequence { } n x is defined by 11 1 = x , 23 2 = x and 2 1 12 − − + = n n n x x x for 3 ≥ n . Prove that ( ) n n n x 3 4 2 − − ⋅ = for all P n ∈ . 5. (8 marks) Prove by induction that n n 1 2 1 ... 3 1 2 1 1 2 2 2 − ≤ + + + + for all positive integers n . You may use, without proof, the fact that ( ) 1 1 1 1 1 2 + − ≤ + k k k for 1 ≥ k . 6. (a) (5 marks) Suppose Z c b a ∈ , , . Prove that if a = b + c , then gcd( a , b ) = gcd( b , c ). (You must prove this directly without using Proposition 2.21, which states that if Z r q b a ∈ , , , with a = qb + r , then gcd( a , b ) = gcd( b , r ).) (b) (4 marks) Suppose P n ∈ . Determine, with proof, all possible values of gcd(8 n + 20, 5 n + 13). Be sure to also show by example that each of these values is possible....
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Fall2006 M1 - 70 266 2016 = − v u with the additional...

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