oops, you're right. I can't figure out what the pattern is. What does he mean by "canceling out"?
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
Not quite the solution you asked for. Close enough?