More 10-Digit Numbers

Source: Original, based on a previous puzzle from Martin Gardner.

Solutions were received from Kirk Bresniker, Joseph DeVincentis, Jozef Hanenberg.  There may be others.  I lost all saved email received prior to 7/12/05. Jozef's summary is below.

The original problem (i.e. k=9) has only one solution: 3816547290
k=0 has only one solution: 8165432709
k=1 has only one solution: 3654729018
k=5 has two solutions: 7290183654 and 7290381654
k=8 has only one solution: 0381654729 (if a leading zero is allowed, otherwise there's no solution) 
k=2, 3, 4, 6 or 7 have no solutions
Notice the similarity between the solutions for k=1 and the first for k=5 and also between the second for k=5 and the solutions for k=8 and k=9
The method used is the following: look first where 0 and 5 must be placed, next look for the places where the even and odd digits must come: they come in an alternating pattern.  Then look for divisibility by 3 or 9, combine it with divisibility by 4 and 8 and finally you look for divisibility by 7.

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