Associated Topics || Dr. Math Home || Search Dr. Math

### 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....

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
```
Associated Topics:
High School Puzzles
Middle School Factoring Numbers
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search