Largest number divisible by its digits

Largest number divisible by its digits is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Find the largest base 10 integer whose digits are all different, that is evenly divisible by each of its individual digits.

For example: 135 is evenly divisible by 1, 3 and 5.

Note that the digit zero (0) can not be in the number as integer division by zero is undefined. The digits must all be unique so a base 10 number will have at most 9 digits.

Feel free to use analytics and clever algorithms to reduce the search space your example needs to visit, but it must do an actual search. (Don't just feed it the answer and verify it is correct.)

Stretch goal

Do the same thing for hexadecimal.

Perl 6

Works with: Rakudo version 2017.08

Base 10

The number can not have a zero in it, that implies that i can not have a 5 either since if it has a 5, it must be divisible by 5, but the only numbers divisible by 5 end in 5 or 0. It can't be zero, and if it is odd, it can't be divisible by 2, 4, 6 or 8. So that leaves 98764321 as possible digits the number can contain. The sum of those 8 digits is not divisible by three so the largest possible integer must use no more than 7 of them. So, strictly by removing possiblities that cannot possibly work we are down to at most 7 digits.

In order to accomodate the most possible digits, the number must be divisible by 7, 8 and 9. If that is true than it is automatically divisable by 2, 3, 4, & 6 as they can all be made from the combinations of multiples of 2 and 3 which are present in 8 & 9. So we'll only bother to check multiples of 9 * 8 * 7 or 504.

All of this optimizations get the run time to well under 1 second.

<lang perl6>my $magic-number = 9*8*7; # 504

my $div = 9876432 div $magic-number * $magic-number; #largest 7 digit multiple of 504 < 9876432

for $div, { $_ - $magic-number } ... 0 -> $test {

   next if $test ~~ / <[05]> /;    # skip numbers containing 0 or 5
   next if $test ~~ / (.).*$0 /;   # skip numbers with non unique digits
   next unless all $test.comb.map: { $test %% $_ }; # skip numbers that don't divide evenly by all digits

   say "Found $test"; # Found a solution, display it
   for $test.comb {
       printf "%s / %s = %s\n", $test, $_, $test / $_;
   }
   exit

}</lang>

Output:

Found 9867312
9867312 / 9 = 1096368
9867312 / 8 = 1233414
9867312 / 6 = 1644552
9867312 / 7 = 1409616
9867312 / 3 = 3289104
9867312 / 1 = 9867312
9867312 / 2 = 4933656

Base 16

There are fewer analytical optimizations available for base 16. Other than 0, no digits can be ruled out so a much larger space must be searched. Rather than using a magic number to try to filter the search space and then check for uniqueness, we'll just generate unique permutations and check them for divisibility. In practice this seems to be much faster. We'll start at the largest possible permutation (FEDCBA987654321) and work down so as soon as we find a solution, we know it is the solution

<lang perl6>my $hex = 'FEDCBA987654321'; # largest possible hex number

my %hex = (flat 1..9, 'A'..'F') Z=> flat 1..15; # pre-calculate hex to decimal values

for $hex.comb.permutations -> @test {

   my $test = [~] @test;
   my $num = $test.parse-base(16);

   next unless all @test.map: { $num %% %hex{$_} };

   say "Found $test"; # Found a solution, display it
   say ' ' x 12, 'In base 16', ' ' x 36, 'In base 10';
   for @test {
       printf "%s / %s = %s  |  %d / %2d = %19d\n",
         $test, $_, ($num / %hex{$_}).base(16), $num, %hex{$_}, $num / %hex{$_};
   }
   exit

}</lang>

Output:

Found FEDCB59726A1348
            In base 16                                    In base 10
FEDCB59726A1348 / F = 10FDA5B4BE4F038  |  1147797065081426760 / 15 =   76519804338761784
FEDCB59726A1348 / E = 1234561D150B83C  |  1147797065081426760 / 14 =   81985504648673340
FEDCB59726A1348 / D = 139AD2E43E0C668  |  1147797065081426760 / 13 =   88292081929340520
FEDCB59726A1348 / C = 153D0F21EDE2C46  |  1147797065081426760 / 12 =   95649755423452230
FEDCB59726A1348 / B = 172B56538F25ED8  |  1147797065081426760 / 11 =  104345187734675160
FEDCB59726A1348 / 5 = 32F8F11E3AED0A8  |  1147797065081426760 /  5 =  229559413016285352
FEDCB59726A1348 / 9 = 1C5169829283B08  |  1147797065081426760 /  9 =  127533007231269640
FEDCB59726A1348 / 7 = 2468AC3A2A17078  |  1147797065081426760 /  7 =  163971009297346680
FEDCB59726A1348 / 2 = 7F6E5ACB93509A4  |  1147797065081426760 /  2 =  573898532540713380
FEDCB59726A1348 / 6 = 2A7A1E43DBC588C  |  1147797065081426760 /  6 =  191299510846904460
FEDCB59726A1348 / A = 197C788F1D76854  |  1147797065081426760 / 10 =  114779706508142676
FEDCB59726A1348 / 1 = FEDCB59726A1348  |  1147797065081426760 /  1 = 1147797065081426760
FEDCB59726A1348 / 3 = 54F43C87B78B118  |  1147797065081426760 /  3 =  382599021693808920
FEDCB59726A1348 / 4 = 3FB72D65C9A84D2  |  1147797065081426760 /  4 =  286949266270356690
FEDCB59726A1348 / 8 = 1FDB96B2E4D4269  |  1147797065081426760 /  8 =  143474633135178345