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MATH135: Algebra

Assignment 3

Fall 2022

Instructions

• There are six submission slots on Crowdmark: Q1, Q2, Q3, Q4, Q5 and Q6. Please upload your solutions into the appropriate slots.

• This assignment covers sections 3.2 - 3.5 (inclusive).

Problems

An integer p > 1 is prime when its only positive divisors are 1 and p. Otherwise, we say that p is composite. Equivalently, p > 1 is composite if it can be written as a product of two integers, both of which are greater than 1.

Q1.  Prove that for all a,b,c ∈ Z, if a | b and a | c, then a | (b + c)2

Q2. Let a,b,c ∈ Z. Consider the implication S(a,b,c): If a ∤ 3b and a | (b + 12c), then a ∤ 9c. (a) State the negation of S(a,b,c). Do not use the word “not”or the negation symbol. (b) State the converse of S(a,b,c).

(c) State the contrapositive of S(a,b,c).

(d) Prove Aa,b,c ∈ Z,S(a,b,c).

Q3.  Prove or disprove the following statements. Be sure to clearly indicate whether the statement

is true or false.

(a) For all x,y ∈ R, if −2x + 5y ≥ 0, then x − y < 0 or −2x2 + 7xy − 5y2  ≥ 0. (b) For all x,y ∈ R, if x − y < 0 or −2x2 + 7xy − 5y2  ≥ 0, then −2x + 5y ≥ 0.

Q4.  Prove that for all a ∈ N, if for all b ∈ Z,a | (6b + 8), then a = 1 or a = 2.  Q5.  Prove that there exists an integer k such that for every natural number x,

x3k+15 + x2 − kx + 5

is a composite integer.

Hint: Consider values of k that simplify the term x3k+15 .

Q6. Let a and b be natural numbers, not both 1. Prove that if a4 +4b is prime, then a is odd and

b is even.

Hint: Note that x4 + 4y4 = [(x − y)2 + y2][(x + y)2 + y2].