thái

Exercise 1.1 . Let with . Show that if and only if . Proof ( q.e.d ). Exercise 1.2 . Let be a composite integer. Show that there exists a prime dividing with . Proof We use strong induction. Let be the proposition that for all composite integer , there exists a prime dividing with . Base case: is true. Inductive step: Now we must show that imply for all composite integer . Since is a composite, there exists so that with (if , we just swap them ). If is a prime, then is true since divides and . If is a composite, then according to the induction assumption, there exists a prime that divides . This prime also divides with . So P(n) is true. Therefore, the claim is true by strong induction for all composite integer n ( q.e.d ). Exercise 1.3 . Let be a positive integer. Show that for every real number , the number of multiples of in the interval is ; in particular, for every integer , the number of multiples of among is . Proof * T...