tye,
I borrowed Algorithm::Loops for the following brute-force approach:

#!/usr/bin/perl
use strict;
use warnings;
use Algorithm::Loops qw/NestedLoops NextPermute/;

my $max = 1;
my $next = GenPowerSet(9);

while ( my @list = $next->() ) {
    PERMUTE:
    while ( NextPermute(@list) ) {
        my $num = join '', @list;
        next PERMUTE if $num < $max;
        for ( @list ) {
            next PERMUTE if $num % $_;
        }
        $max = $num if $num > $max;
    }
}
print $max;

sub GenPowerSet {
    my $end = shift;
    return NestedLoops(
        [
            [ 1..$end ],
            ( sub {
                [ $_+1 .. $end ]
            } ) x $end,
        ],
        { OnlyWhen => 1, },
    );
}
[download]

It came up with 9_867_312 in about 30 seconds.