Mathematics 243, section 1 – Algebraic Structures Makeup Exam 2, November 3, 2006

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}) ⇔ x_{1} − 3y_{1} = x_{2} − 3y_{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^{2} + 3^{2} + 5^{2} + ⋯ + (2n − 1)^{2} = \frac{n(2n − 1)(2n + 1)}{3}.

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(456, 120) and express d in the form d = m · 456 + n · 120 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 17x ≡ 5 mod 32 with 0 ≤ x < 31.

B) (10) For which [b] ∈ Z_{15} do solutions [x] ∈ Z_{15} of the equation [6][x] = [b] exist? Explain how you can tell.