Mathematics 243, section 1 – Algebraic Structures Exam 2, November 1, 2006

Directions:

Do all work in the blue exam booklet. There are 100 possible points.

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.

IV. (10) Let a, b, c be integers. Show that if gcd(a, b) = 1 and a|(bc), then a|c.

V. A) (10) Find a solution x of the congruence 31x ≡ 2 mod 64 with 0 ≤ x < 63.

B) (10) For which [b] ∈ Z_{16} do solutions [x] ∈ Z_{16} of the equation [12][x] = [b] exist? Explain how you can tell.