1) In this problem, you may use the fact (which we will prove in Chapter 6)...

1) In this problem, you may use the fact (which we will prove in Chapter 6) that an integer n is not divisible by 3 if and only if there exists an integer k such that n = 3k + 1 or n = 3k + 2.

(a) Prove that for all integers n, if 3 | n2, then 3 | n.

2) Let a and b be positive integers. Prove that if a | b and b | a, then a = b.

3) Determine whether each statement is true or false. If true, then prove it. If false, then provide a counterexample.

(a) The sum of two irrational numbers is irrational.

(c) The product of a nonzero rational number and an irrational number is irrational.

Expert Solution

Colby Messinger answered 2 years ago

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