Pow Divisor Counting

Your task: To figure out as much as ou can about how many divisors a number has

you will feel like you have more direction if you have certain questions to investigate:

1. What kinds of numbers have exactly 3 divisors?

2. What kind of numbers have exactly 4 divisors?

3. What is the smallest number that has 20 divisors?

4. Do bigger #s necessarily have more divisors?

5. Is there a way to figure out how many divisors 1,000,000 has without actually doing the listing and counting? How about one billion?

6. What kind of numbers have exactly 100 divisors?

7. How many have 7?

Process: When I looked at my list and the list from the homework Prime Time, I noticed a majority of the numbers with 3 divisors were square numbers. For example: 4, 9, 25 etc. But it’s not just square numbers. 81 is a square number but has more than 3 divisors. With 4, 9, and 25 I tried to write the scientific notation for them. For 4 it was 22. 9 was 32 and 25 was 52. What they all have in common is that they are all prime numbers to the squared power. To make sure that was correct, I tried 112. I got 121, which only has 3 divisors: 1, 11, 121.

A number with 3 divisors are prime number squared. After, I saw that 28 had 3 divisors but was not only prime squared but multiplied by 7. This happens again with the number 18, which is 32 x 2. This means that not only are they prime numbers squared but they are also prime number squared but could also be multiplied by another prime number.

With this pattern discovered, I tried to see if it continued into other numbers. I got numbers with 4 divisors and also saw that instead of a prime number squared, they were prime number cubed! For instance, 8 is 23 and 27 is 33. They all had 4 divisors and they were prime number cubed. To check I did 5 as the next prime number and cubed it. It came out as 125 which has 4 divisors: 1, 125, 25 and 5. This shows that a number with 4 divisors are prime number cubed. Before I moved on I noticed something else. The number 6 had 4 divisors but was not a prime number cubed, it is 2x3 which