Prime Ponderings
Math brain teasers require computations to solve.
What are the largest and smallest 5digit numbers that satisfy the following conditions?
1. Each digit of the number is a prime digit.
2. Each successive pair of digits forms a 2digit number that is NOT a prime number.
3. Each of the prime digits must appear at least once in the 5digit number.
Answer
Largest  35772
Smallest  32257
Prime digits are 2, 3, 5 and 7. Taken as pairs, the only combinations that fit criterion 2 are:
22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75 and 77.
Of that list, 33 is the only number that contains a 3 as the second digit and so for criterion 3 to be satisfied, 3 can only appear at the start of the 5digit numbers. From there it is simply a matter of choosing the next digits using the list above so that criterion 2 is satisfied, making sure that criterion 3 is also satisfied.
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Comments
jimrcook
Dec 16, 2004
 Very nice one. I'd probably have solved it better if I hadn't misread criterion 3. I thought it said each must appear only once, and decided you must have counted 1 as a prime digit. There's no solution to that one. Sigh. Once I got over that, I saw the answer. 
(user deleted)
Jan 02, 2005
 Great puzzle. I enjoyed it. 
cloud_strife
Jan 19, 2005
 "Of that list, 33 is the only number that contains a 3 as the second digit and so for criterion 3 to be satisfied, 3 can only appear at the start of the 5digit numbers" hey, i don't get this part of the answer.. can u explain it more clearly? thanks 
brianz
May 11, 2005
 Good one. Very fun. 
solidtanker
Jun 24, 2005
 Enjoyable. I love prime numbers, they are a great mystery! 
Sunrose
Sep 11, 2005
 I enjoyed this math teaser
Good Job 
javaguru
Jan 28, 2009
 It was easier to list the numbers that can,t be formed (23, 37, 53, 73) than the ones that can. This made the requirement of starting with a 3 clear and the rest was easy.

spikethru4
Feb 10, 2011
 @jimrcook  I made the same mistake, but if you include 1 as a 'prime digit' the smallest is 32157 and the largest 35721. 
eddandi
Feb 16, 2011
 1 is a prime number and is not included in this. why? 
javaguru
Mar 09, 2011
 By definition, 1 is not a prime number. 
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