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MATH 301 – HOMEWORK VII


1. Solve the inequality | x+1 | < | x2 - 1|, where x is a real number.


2.

a. Show that if a is an integer, then there exists an integer k such that a = 3k + r where r = 0, 1, or 2.

b. Let a and b be positive integers. Show that if 3 divides the product ab, then either 3 divides a or 3 divides b.

c. Show that if a positive integer n is not divisible by 3, then n2 - 1 is divisible by 3.


3. The logarithm ln(x) has the property that ln(xy) = ln(x)+ln(y) for all positive real numbers x and y. Show that ln(xr )=r ln(x) for all positive numbers x and all rational numbers r.


4. Provide a counterexample to each of the following:

a. The sum of two irrational numbers is irrational.

b. If a and b are integers such that ab is a perfect square, then a and b are perfect squares.