Chapter 18, Problem 25E

Assume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers .

25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)

Let a and b be integers such that ab and ba. Prove that b=0.

Use the second principle of Finite Induction to prove that every positive integer n can be expressed in the form n=c0+c13+c232+...+cj13j1+cj3j, where j is a nonnegative integer, ci0,1,2 for all ij, and cj1,2.

Let (a,b)=1. Prove that (a,bn)=1 for all positive integers n.

Let a be an integer. Prove that 3|a(a+1)(a+2). (Hint: Consider three cases.)