Question: What patterns do the divisors of
the positive integers make?
The numbers in the set {1, 2, 3, 4, 5, . . .} are the positive integers. The divisors of
an integer are those integers that divide the given integer evenly, that is,
with zero remainder. The set of divisors of the integer n is denoted Dn.
1. Make a table of divisors for the integers
from 1 through 30.
If the integer itself is excluded from the
set of its divisors, the new set is called the set of proper divisors of
the integer. It is denoted PDn.
2. Add a column to your table in #1 of the
proper divisors of each integer in the table.
3. List all the numbers in your table for
which PDn contains only the number
1. Those integers greater than 1 in this list are called prime numbers. State
another definition of prime numbers which uses the set of divisors Dn.
4. A composite number is an integer greater
than one that is not prime. List the composite numbers
from 1 to 30. State a definition of composite numbers that uses the set of
divisors Dn.
5. Every composite number can be factored
into a product of primes in exactly one way. Add a column to your table in #1
of the prime factorization of each integer in the table.
6. Find the sum of the proper divisors of
each integer in table #1. How does the sum compare to each corresponding
integer?
An integer is abundant if the sum of
its proper divisors is greater than the integer. It is deficient if the
sum of its proper divisors is less than the integer. An integer is perfect
if the sum of its proper divisors is equal to the integer.
7. Identify each of the integers from 1 to 30
as abundant, deficient, or perfect.
8. Compute the following multiples of the perfect
number 6 and determine whether each is abundant, perfect, or deficient.
7×6, 8×6, 11×6, 15×6