Homework:

1. Read Chapter 15 and complete Exercises 15.2, 15.3, 15.4, 15.7
2. Complete the Exercises given below.

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.

1. Prove that is a multiplicative function by filling in the sketch of the proof provided below.

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.

Hence .

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 Euler’s Theorem.

Theorem (Euler). Every even perfect number is of the form where both p and are prime.

Proof.

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 .