1. (6 marks) Prove that for any m Є Z, m² = 5k or m² = 5k 1 or m² = 5k + 4 for some k Є Z. 2. (6 marks) Let p be a prime. (a) Prove that √p is irrational. (b) Complete the statement: Vn Є Z+, √√n is irrational if and only if. _. Prove it. 3. (6 marks) Use the Euclidean algorithm to find an integer solution to the equations: (a) 1053x+481y = 13 (b) 3094x+2513y = 7 a1 4. (6 marks) Let a = p₁₁pan and b = p₁₁pon be the prime factorizations of integers a and b, where some of the exponents maybe zero. Prove the following: (a) gcd(a,b) = p₁₁ P1 d₁... di pd, where d; = min(a, b) for all 1 ≤ i ≤n. (b) lcm(a,b) =pp, where l₁ = max(a;, b;) for all 1 ≤i≤n. (c) a b = gcd(a, b) lcm(a, b). 5. (6 marks) li (a) Prove that the equation ax + by = c has an integer solution if and only if gcd(a, b)|c. (b) Find all integer solutions to each of the equations: (i) 28x+35y = 60 (ii) 21x+15y = 12 6. (6 marks) Given a, b, c = Z+ (a) Propose a definition for gcd(a, b, c). (b) Prove that gcd(a, b, c) = gcd(a, gcd(b, c)). (c) Prove that the equation ax + by + cz = e has an integer solution if and only if gcd(a,b,c)|e.

1. (6 marks) Prove that for any m Є Z, m² = 5k or m² = 5k 1 or m² = 5k + 4 for some k Є Z. 2. (6 marks) Let p be a prime. (a) Prove that √p is irrational. (b) Complete the statement: Vn Є Z+, √√n is irrational if and only if. _. Prove it. 3. (6 marks) Use the Euclidean algorithm to find an integer solution to the equations: (a) 1053x+481y = 13 (b) 3094x+2513y = 7 a1 4. (6 marks) Let a = p₁₁pan and b = p₁₁pon be the prime factorizations of integers a and b, where some of the exponents maybe zero. Prove the following: (a) gcd(a,b) = p₁₁ P1 d₁... di pd, where d; = min(a, b) for all 1 ≤ i ≤n. (b) lcm(a,b) =pp, where l₁ = max(a;, b;) for all 1 ≤i≤n. (c) a b = gcd(a, b) lcm(a, b). 5. (6 marks) li (a) Prove that the equation ax + by = c has an integer solution if and only if gcd(a, b)|c. (b) Find all integer solutions to each of the equations: (i) 28x+35y = 60 (ii) 21x+15y = 12 6. (6 marks) Given a, b, c = Z+ (a) Propose a definition for gcd(a, b, c). (b) Prove that gcd(a, b, c) = gcd(a, gcd(b, c)). (c) Prove that the equation ax + by + cz = e has an integer solution if and only if gcd(a,b,c)|e.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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