I will ask for volunteers to present Exercises 1 and 2 (below) in class next lesson.† I will collect Exercise 3 on Thursday, November 10.†
Theorem.† If gcd(m, n) = 1, then .
††††† Sketch of Proof. †
a) Assume that m and n are two relatively prime positive integers, and prove that any divisor d of mn can be written uniquely as a product d1d2 where (d1, d2) = 1 and d1|m, d2|n.
b) Conversely, every pair of divisors d1|m, d2|n satisfying (d1, d2) = 1 gives rise to a divisor d1d2 of mn.
2.†† a) Prove that if f is a multiplicative function and n has prime factorization , then
b) Deduce that if n has prime factorization , then
3.†† Work through parts a) through d) below to prove Eulerís Theorem.
Theorem (Euler).† Every even perfect number is of the form where both p and †are prime.
a) Suppose that n is an even perfect number and write , where s is odd.† Show that †and .
b) Let ††Show that if t > 1, then ; that is, .
c) Derive a contradiction from the fact that t > 1; hence t = 1 and ††Conclude that .
d) Explain why part c) implies that prime.† Conclude that r +1 must be prime too.† Hence .