Prime Number Puzzle

Date: 09/27/2001 at 20:56:08
From: Pat
Subject: Prime number word problem

I am the product of four primes, and the sum of these numbers is 
thirty. My three digits are prime and differant. Who am I?

I couldn't come up with any combination without using one (1,5,11,13 
for 715) How do I do this? Can I use a prime number twice in my set, 
like 2,2,3,3,7,13 because I am using only four primes even though I am 
using them more then once? 

Date: 09/28/2001 at 14:12:39
From: Doctor Ian
Subject: Re: Prime number word problem

Hi Pat,

The main thing here is to proceed systematically. For example, you 
know that you're looking for a 3-digit number, with distinct prime 
digits. How many of those are there? Not very many.

The digits have to be 2, 3, 5, or 7, and they can't repeat, so you 
need to find all the permutations of the 3-element subsets of this 
set:

  2--  =>  23-  =>  235
                    237

           25   =>  253
                    257

           27-  =>  273
                    275

  3--  =>  32-
           35-
           37-

  5--  =>  52-
           53-
           57-

  7--  =>  72-
           73-
           75-

I'll leave it for you to fill in the rest of the table. The number of 
possibilities is 4*3*2 = 24, which isn't too awful.  

Once you have the 24 numbers, you can break them into prime factors.  
Keep the ones that have four prime factors; and of those, keep the 
ones whose factors add up to 30. Maybe there will be only one of 
those, maybe not. 

By the way, note that 1 is _not_ a prime number, so it won't appear as 
a digit in the number, or as one of the prime factors.  

In case you don't feel like working out the prime factorization of 24 
different 3-digit numbers, you can look them up here:

Prime Factorization - Cenius.net
http://www.cenius.fsnet.co.uk/refer/maths/articles/p/primefactorization1000_ref.html   

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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