Problem 1. For integers means there exists an integer q such that qa = b. This predicate defines a relation on a and b, let a|b be a predicate denoting "a divides b." Formally, alb the set of all integers. Prove that this relation is transitive. Problem 2. Suppose b1 has a remainder ri when divided by a, and b2 has a remainder r2 when divided by when divided by a. You can use the fact that the remainder exists and is unique a. Prove (in paragraph form) that bi +b2 has a remainder r1 + r2 or ri+r2- a.
Problem 3. For any real number c, define [c] - read as ceiling - as an integer d such that 3z with 0 < z < 1 and c = d - z. Assume, as a premise, that c] exists for every real c.
A) Prove that for any c, b)
B) Prove that Va, b : [a + b] is equal to alb a1 b] 1 or -
Problem 4. Prove that Vn E N, 2 = 2"1-1 (in other words, 1+2+4+8+..+ 2" 2n+1- 1). Use the following proof strategy: Proof by contradiction. Take the set of all n for which this statement is false. Use the fact that every set of non-negative integers has a minimum Call that minimum k. Either k = 0 or the statement is true for k - 1. Get a contradiction from here.