Update: Here's a solution:

use Math::BigInt       ();
use Math::Big::Factors ();

my $a = Math::BigInt->new(1);
$a *= $_ foreach @a;  # 17820000

my $b = Math::BigInt->new(1);
$b *= $_ foreach @b;  # 19872000

my $gcd = Math::BigInt::bgcd($a, $b);  # 108000

my $c = $a / $gcd;  # 165
my $d = $b / $gcd;  # 184

my @c = Math::Big::Factors::factors_wheel($c);  # 3, 5, 11
my @d = Math::Big::Factors::factors_wheel($d);  # 2, 2, 2, 23
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Not quite the solution you asked for. Close enough?