16 CHAPTER 2 Basic Techniques to Prove If/Then Statements (Continued ) If we now set r = a - qb, we can show that this number satisfies the conditions listed in the conclusion. By the previous inequalities (*), 0 < r < b. By its definition, a = qb + r. Since the two cases presented cover all the possibilities, we proved that the statement is true. The statement in Example 2.6 is part of the theorem known as the division algorithm. The beginning of the proof uses an idea that, while clever, might not look intuitive. But, on the other hand, it does not use advanced mathematical concepts. It is an example of how the combination of basic ideas can yield more advanced results. Later on we will prove that the numbers q and r we just found are the only ones satisfying the required properties (see the exercises at the end of the section on uniqueness theorems). To be precise, Example 2.6 could have been included in the section on existence theorems, because it states that there exist two numbers having certain properties. Moreover, its proof is constructed by considering two sepa- rate cases. Thus, one could argue for its inclusion in the section on multiple hypotheses! Exercises Rewrite the following statements in the form "If/then." Then prove them. 11. The average of two consecutive odd integers is always even.