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### 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
website is great. Keep up the great work.

Thanks again,
Molly McS.
```
Associated Topics:
High School Number Theory
High School Puzzles
Middle School Prime Numbers
Middle School Puzzles

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