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2007 F Midterm - u ≤ 2000 and 0 ≤ v ≤ 2000 4 Suppose...

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MATH 135 Fall 2007 Midterm #1 Monday 15 October 2007, 7:00 p.m. to 8:15 p.m. 1. Let a, b, c Z . Suppose that P is the statement: “If a | b or a | c , then a | bc ”. (a) Write down the converse of P . [2] (b) Disprove the converse of P . [2] (c) Write down the contrapositive of P . [2] 2. Suppose that x 1 = 10, x 2 = 16 and x n = 4 x n - 1 - 4 x n - 2 for n 3. [8] Prove that x n = 3(2 n +1 ) - n 2 n for all n P . 3. (a) Determine the complete solution to the linear Diophantine equation 1357 x + 1110 y = 8. [7] (b) Write down the complete solution to the linear Diophantine equation 1357 u - 1110 v = 8. [3] (No justification is necessary in this part.) (c) Determine the number of solutions to the linear Diophantine equation 1357 u - 1110 v = 8 [4] with 0
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Unformatted text preview: u ≤ 2000 and 0 ≤ v ≤ 2000. 4. Suppose m, n ∈ Z . Prove that if m | n and n 6 = 0, then | m | ≤ | n | . [4] 5. Suppose that e, f, g ∈ Z . Prove that if the linear Diophantine equation ex + fy = g has an [4] integer solution, then gcd( e, f ) | g . 6. (a) Suppose that a, b, c ∈ P . Prove that if gcd( ac, b ) = 1, then gcd( a, b ) = 1. [4] (b) Suppose that a, b, d ∈ P . Prove that if gcd( a, b ) = 1 and d | a , then gcd( a, db ) = d . [4] 7. Suppose that x, y ∈ R with x > y > 0. [6] Prove by induction on n that ( x + y ) n ≥ x n +1-y n +1 x-y for all n ∈ P ....
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