Mine is very slow, but short. I wrote it before reading the rest of the thread. I figured that we knew the answer had at most 8 characters, as it couldn't have both 2 and 5, so I counted down from the largest 8-different-digit number, guessing the answer would be closer to the top than the bottom. I realized that checking both 9, 8, 6 and their divisors was a waste, but guessed that the logic to exclude those would outweigh the extra modulos.

LOOP: for (my $i = 98765432; $i > 0; $i--) {
  next if $i =~ /0/ or $i =~ /(\d)\d*\1/;
  $i % $_ and next LOOP for split '', $i;
  print "$i\n";
  exit;
}
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Updated: Re-worded the last sentence to correct a word-processing-o.