Mathematics 243, section 2 – Algebraic Structures Makeup Exam 2, November 5, 2012

I. Think of R^{2} (the set of ordered pairs of real numbers) as the ordinary Cartesian coordinate plane. Let R be the relation on R^{2} defined by

(x_{1}, y_{1})R(x_{2}, y_{2}) ⇔ 3x_{1} + y_{1} = 3x_{2} + y_{2}

A) (15) Show that R is an equivalence relation on R^{2}.

B) (5) Draw a picture of the equivalence class [(2, 3)] for this relation.

II. (20) Prove by mathematical induction: For all n ≥ 1,

1^{3} + 3^{3} + 5^{3} + ⋯ + (2n − 1)^{3} = n^{2}(2n^{2} − 1).

III. A) (15) Give the precise statement of the division algorithm in Z, and prove the existence

part.

B) (15) Use the Euclidean algorithm to find the integer d = gcd(576, 99) and express d

in the form d = m · 576 + n · 99 for some integers m, n. C) (15) Find all solutions x ∈ Z of the congruence 27x ≡ 2 mod 64.

IV. (15) Let a, b, c be integers. Show that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1.

Extra Credit (10) Let a_{1} = 0, a_{2} = −30 and a_{n} = 8a_{n−1} − 15a_{n−2} for all n ≥ 2. Show that a_{n} = 5 · 3^{n} − 3 · 5^{n} for all n ≥ 1.