Hit-and-Run Prime Number Problem

Date: 12/10/2002 at 21:41:36
From: Charlie
Subject: Hit-and-Run Prime Number Word Problem

Cop: Did You get his license number?

Victim: Yes! His license was in two parts, a two-digit number and a 
three-digit number. The two-digit number was prime and the sum of 
the two digits was a two-digit prime. The tens digit was larger than 
the units digit. In the three-digit part, the digits were all odd 
and different. The sum of the three digits was palindromic. The sum 
of the first and third digits was one-half the sum of the first and 
second.  That's all I remember!

Please help - I can't figure this out.

Date: 12/11/2002 at 11:00:10
From: Doctor Ian
Subject: Re: Hit-and-Run Prime Number Word Problem

Hi Charlie,

There aren't all that many two-digit primes:

     11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 
     53, 59, 62, 67, 71, 73, 79, 83, 89, 97

We're told that the tens digit is larger than the units digit, so that
eliminates some possibilities and we're left with:

     31, 41, 43, 
     53, 62, 71, 73, 83, 97

The sum of the digits is palindromic, so it has to be two digits, 
which rules out some more possibilities, and we have:
 
     73, 83, 97

In only one of these cases is the sum a prime number. So that tells
you the first number in the license plate. 

You need to do the same kind of thing to find the second number. Start 
with all the possibilities, and use the clues to eliminate the ones 
that aren't consistent with the clue. When you're done, only one
possibility should remain. 

Can you take it from here? 

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