# Thread: A Proper Factor Issue

1. ## A Proper Factor Issue

What is the smallest number that contains precisely 18 proper factors?
(A proper factor is a factor other than 1 or the number itself. As an example, the number 12 has four proper factors: 2, 3, 4 and 6)

2. ## Re: A Proper Factor Issue

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>432

Each positive integer > 1 can be written as a product of powers of mutually distinct prime numbers. For example, 9 = 3^2 and 12 = 2^2 * 3^1 and 72 = 2^3 * 3^2.

The total number of factors (including 1 and the number itself) of p^n where p is prime, is n+1. For example, 9 = 3^2 is divisible by 2+1 = 3 numbers: 1, 3 and 9. And 8 = 2^3 is divisible by 3+1 = 4 numbers: 1, 2, 4 and 8.

The total number of factors of an arbitrary number is the product of the number of factors of its "power of prime" components. For example, 12 = 2^2 * 3^1. The number 2^2 has 2+1 = 3 factors, and the number 3^1 has 1+1 = 2 factors, so 12 has 3*2 = 6 factors, of which 6-2 =4 are proper factors.

We are looking for a number with 18 proper factors, so 20 total factors. The most "economical" way of factoring 20 is 5*4; this corresponds to p^4 * q^3 where p and q are prime numbers. The smallest primes are p=2 and q=3, leafing to 2^4 * 3^3 = 432.</font color=yellow></span hi>

3. ## Re: A Proper Factor Issue

For the moment, I'm going to chalk this puzzle up as "unsolved"...
as I think I have a better answer.

But, I'll leave room for the possibility that Hans may be on to something. <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

4. ## Re: A Proper Factor Issue

How about <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>240</font color=yellow></span hi>

5. ## Re: A Proper Factor Issue

Yep, that's it.

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>20 = 5 * 2 * 2, corresponding to p^4 * q^1 * r^1, take p = 2, q = 3, r = 5</font color=yellow></span hi>

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