Theorem. Let n e Nand let a E Z. If a = ng +r and 0

a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self: write out what "if and only if' means. You'll be proving two conditional statements. (b) Write a new theorem: Let a be an integer with a > 0. The integer a is threeven if and only if = 3k.) (Definition of threeven an integer n is called threeven if there exists an integer k so that n =

6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the even integers 0, 2, 4, 6, ..., 2m-2.

Let K be an integer satisfying 1 ≤ K ≤ N. In how many ways can we arrange the N objects in B so that B[K] is smaller than all the objects that follow it in B (i.e., B[K] < B[I] for all subscripts I greater than K)? Answer the question for N = 6, K = 4 first. Then give the answer for general N and K. Simplify your answer algebraically as much as possible.