1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k. 2. Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or (an)n=a.

4. Let a 0 in the ring of integers Z. Find b = Z such that ab but (a) = (b).

Consider F = {a+b√√/2 : a, b = Q}, equipped with the usual addition "+" and multiplication "." as real numbers. Show that F is a field. 99

Let S denote the set of positive real numbers. Define operations and by the equations x + y =xy (a) Prove that S with the operations and (b) Is S commutative? and x Oy:= x¹ny is a ring. What are 0s and 1s? (c) Find the units and the zero divisors in S. For every unit u € S, find its inverse v, i.e., v € S such that u v = vu = 1s.