MATH 301 – HOMEWORK VII
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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.
2021-12-23