In order to compute

_{}(n) for values ofnthat are more complicated than prime powers, we would like to develop a means for breaking the general computation into several simpler computations. One way to proceed is to determine combinations ofmandnfor which the statement_{}(m·n) =_{}(m)·_{}(n)is true. This leads us directly to the next Research Question.

## Research Question 3

Find a condition for pairs of integers

mandnwhich guarantees that_{}(mn) =_{}(m)_{}(n).

Note:The proof of your conjecture (assuming you find the conjecture we are looking for) will be considered on your homework assignment. You can provide a proof with the lab report if you have one, but this is not required. However, since you are not mandated to include a proof for your conjecture, you should providelotsof numerical data to support your claim.

## Research Question 4

Use your conjectures from Research Questions 2 and 3 to assist in finding a formula for

_{}(n), wheren=p,^{a}q^{b}pandqare distinct primes, andaandbare positive integers.We're almost there. Here's the last step:

## Research Question 5

Find a formula for

_{}(n), wheren=_{},

p_{1},p_{2}, . . . ,p_{k}are distinct primes, anda_{1},a_{2}, . . . ,a_{k}are positive integers.

## Exercise 2

Use your formula from Research Question 5 to compute

_{}(10^{20}), explaining the steps. Compare the result with your answer to Exercise 1(c).

Section 9.1 | Section 9.2 | Section 9.3 | Section 9.4

Copyright © 2001 by W. H. Freeman and Company