**Homework: **

**Read Chapter 15 and complete Exercises 15.2, 15.3, 15.4, 15.7****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.
**

- 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 *d*_{1}*d*_{2}
where (*d*_{1}, *d*_{2}) = 1 and *d*_{1}|*m*,
*d*_{2}|*n*.

b) Conversely, every pair of
divisors *d*_{1}|*m*, *d*_{2}|*n*
satisfying (*d*_{1}, *d*_{2}) = 1 gives rise to a
divisor *d*_{1}*d*_{2} 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 _{}.