Sums of Sets of Prime Numbers

Date: 01/07/2003 at 21:56:20
From: Molly 
Subject: High school puzzle

Given several sets of prime numbers, use each of the nine non-zero 
digits exactly once. What is the smallest possible sum such a set 
could have?

An example would be the set: {5, 29,43, 67, 81}. Their sum is 225 but 
this is not the smallest sum. The smallest sum I was able to find was 
207 using the set {61, 47, 2, 89, 35}. I am told this is not correct.

Date: 01/08/2003 at 14:07:18
From: Doctor Ian
Subject: Re: High school puzzle

Hi Molly, 

The first thing that occurs to me is to get the prime digits alone by
themselves:

  {2, 3, 5, 7}    1, 4, 6, 8, 9

Now what can we do with the remaining digits?  41 is prime, and so is 
89:

  {2, 3, 5, 7, 41, 89}    6

6 isn't prime, so maybe we can attach it to another number. Can you
see how to do that? 

But is this the _smallest_ possible sum? Well, let's consider a few
things. There is no way that 4, 6, or 8 can appear in the unit place
of a prime number. (Do you see why?) So those are all going to have
to appear as parts of combined numbers.  

9 can't appear by itself; and right now we have it in the units
position, so we can only make things worse by doing anything else with
it. So are there any changes that you could make to this set that 
would make the sum smaller? 

Does this help?

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

Date: 01/08/2003 at 21:23:35
From: Molly 
Subject: Thank you (High school puzzle)

Thank you for taking the time to answer my question. I greatly 
appreciate it! I found your hints to be very helpful. I think your 
website is great. Keep up the great work. 

Thanks again,
Molly McS.