Numbers with 5 Factors

Date: 02/06/2002 at 21:45:17
From: Kyle
Subject: Numbers with 5 factors

I am trying to find 3 numbers with 5 factors.  I know that 16 has 5 
(1,2,4,8,16). What other two numbers might have 5 factors? Is there a 
way to easily calculate these factors?

Any help would be appreciated.

Thanks, 
Kyle

Date: 02/07/2002 at 10:41:39
From: Doctor Paul
Subject: Re: Numbers with 5 factors

Why don't you think about why 16 works and try to construct other 
numbers in a similar manner?

Notice that a number will have an odd number of factors if and only if 
it is a perfect square (this is the only way that you can have one of 
the factors only count once).

So if you're not considering the perfect squares, then you're 
searching in the wrong place.

Now what's special about 16? It's the 4th power of 2 and is hence 
divisible by 2^0, 2^1, 2^2, 2^3, and 2^4. I think you'll see that p^4 
will have five factors for *any* prime number p.

The factors will be 1, p, p^2, p^3, and p^4.

The next such numbers are 3^4 = 81 and 5^4 = 625

What if p isn't prime? Suppose p = 6 = 2*3. This isn't going to work.

6^4 = 2^4 * 3^4 will have 25 divisors:

1, 2, 2^2, 2^3, 2^4, 3, 3^2, 3^3, 3^4, 2*3, 2*3^2, 3*3^3, 2*3^4, 
2^2*3, 2^2*3^2, 2^2*3^3, etc....

I hope this helps. Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/07/2002 at 11:12:45
From: Kyle
Subject: Numbers with 5 factors

Thank you, Dr. Paul! I think I have it figured out. I was not looking 
at the "perfect square" picture. Now that you have pointed that out, 
it makes total sense to me. I thought I had tried 81 last night, but I 
must have skipped it. My mind sometimes gets on one track and I have 
difficulty thinking outside the box. Thank you for your kind 
assistance!

Kyle