Imp 2 Pow 3: Divisor Counting

IMP 2 POW 3: Divisor Counting.

I. Problem statement:

This POW is all about finding information and patterns about the way divisors of certain numbers are found and expressed. In this POW when we talk about divisors we usually are counting the number of divisors that a number has. The divisor is a number that a number can be divided by, of course every number is divisible by every other number but in these problems we are only talking about whole, positive numbers. Every number is divisible evenly by one or itself so every number has at least 2 divisors. Numbers that have only 2 divisors are called prime numbers, only uneven numbers can be prime numbers, all even numbers except two have at least 3 divisors, the number itself, 1 and 2.

the task was to find information about different questions about divisors, to find patterns and to make our own question.

II. Questions:

What kinds of numbers have exactly 3 divisors?, 4 divisors? and so on.

Do bigger numbers necessarily have more divisors?

Is there a way to figure out how many divisors one million has, how many one billion has and so on without actually counting all the different ones? (in other words is there a system for counting divisors)

What is the smallest number with 20 divisors?

Choose your own interesting question: Do all the rules for divisors work the same with negative numbers?

III. Information gathering

For most of the questions I found the divisors or made generalizations based on other information I gained via other numbers divisors or I used logic to figure out simple rules to apply to the numbers.

I gathered many different simple rules and the number of divisors in some numbers, and my brother showed me the equation to find any numbers divisor. (It wasn’t at all useful because I have no idea what most of the stuff in the equation means and I can’t write it into Word.)

When I found information that I knew was correct by either finding consistent or obvious patterns or by realizing it could be no other way.

VI. Results and conjectures

a.

1.what kinds of numbers have exactly 3 divisors?, 4 divisors?

As a general pattern for 3 divisors all of them have been perfect squares however all perfect squares do not have 3 divisors

b. all had uneven numbers as divisors,

2.Do bigger numbers necessarily have more divisors?

No, because all prime numbers can only have 2 divisors while smaller composite numbers can have more divisors. Also large composite numbers can have less divisors than smaller composite number for example 94 has 4 divisors, 2 and 47, while 60 has 11: 1, 2, 3, 4, 5, 6, 10, 12, 20, 30, 60

3. 1,000,000 has 49 divisors: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160,

200, 250, 320, 400, 500, 625, 800, 1000, 1250, 1600, 2000, 2500, 3125,

4000, 5000, 6250, 8000, 10000, 12500, 15625, 20000, 25000, 31250,

40000, 50000, 62500, 100000, 125000, 200000, 250000, 500000, 1000000 while I did not find a way of counting divisors without counting I did find as many as I could.

1,000,000,000 has 100 divisors: 1, 2, 4, 5, 8,