Problems & Puzzles: Puzzles

Puzzle 698 Others as 129

From the entry 1156 of the always interesting site of Claudio Meller, we generate the following extension:

Given a zero free number N composed by D distinct digits, obtain all of his D! permutations. Calculate the sum S of all of the D!(D!-1)/2 absolute differences between each two permutations, and report N only if N divides S.

I have obtained the following solutions:

 D N S 3 129 172 258 387 6192 4644 4644 3870 4 1287 6981 8592 921492 921492 850608 6 312954 462153 71541284400 50078899080

Q1. Obtain the solutions for D=7, 8 & 9. (BTW, confirm that there are NO solutions for D=2 & 5).

Q2. Is there a N prime solution or this is impossible?

Contributions came from Giovanni Resta, Abhiram R Devesh and Emmanuel Vantieghem

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Resta, Devesh & Vantieghem made the same contribution:

The only solution missing from the table of Puzzle 698 is, for D=7, N=1943568 (S=37148697938880). No solutions for D=8 or D=9.

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