Ex 3. Consider the following definitions:

Definition: Let a and b be integers. A linear combination of aand b is an expression of the form ax + by, where x and y are alsointegers. Note that a linear combination of a and b is also aninteger.

Definition: Given two integers a and b we say that a divides b,and we write a|b, if there exists an integer k such that b = ka.Moreover, we write a - b if a does not divide b.

For each proof state clearly which technique you used (directproof, proof by contrapositive, proof by contradiction). Even ifyou are not able to prove some of the following claims, you canstill use them in the proof of the following ones, if needed.

(a) Given the above definition, is it true that a|0 for all a inZ? Is it true that 0|a for all a in Z? Is it true that a|a for alla in Z? Explain your answers.

(b) Prove that if a and b are two integers such that b?0 anda|b, then |a| ? |b|.

(c) Prove that if a, b and c are three integers such that c|aand c|b then c divides any linear combination of a and b.

(d) Let a be a natural number and b be an integer. If a|(b + 1)and a|(b ? 1), then a = 1 or a = 2. (Hint: you may use a cleverlinear combination...)

(e) Prove that if a and b are two integers with a ? 2, then a -b or a - b + 1