BrowserUk,
The order and pairings you choose to cancel out affect the result. This may prevent optimizations that seek to cancel out terms in a different order (such as largest terms first). For instance in your example, it took 6 cancelings to get the resut. Here is another possibility that doesn't result in prime factorization but only takes 3 pairings:
# After duplicates have been removed
# @a = ( 20, 33, 60 );
# @b = ( 2, 5, 12, 16, 23 );
60 / 12 = 5 / 1
# @a = ( 20, 33, 5 );
# @b = ( 2, 5, 16, 23 );
20 / 5 = 4 / 1
# @a = ( 4, 33, 5 );
# @b = ( 2, 16, 23 );
16 / 4 = 4 / 1
# @a = ( 33, 5 );
# @b = ( 2, 4, 23 );
My algorithm doesn't require prime factorization (just GCD).
#!/usr/bin/perl
use strict;
use warnings;
use Inline 'C';
my (%set_a, %set_b);
++$set_a{$_} for 10, 20, 33, 45, 60;
++$set_b{$_} for 2, 5, 10, 12, 16, 23, 45;
cancel_out(); # Assume %set_a and %set_b at package scope
print "$_\t$set_a{$_}\n" for keys %set_a;
print "\n\n\n";
print "$_\t$set_b{$_}\n" for keys %set_b;
sub cancel_out {
my $finished;
while ( ! $finished ) {
$finished = 1;
for ( keys %set_a ) {
if ( exists $set_b{$_} ) {
my $res = $set_a{$_} <=> $set_b{$_};
if ( ! $res ) {
delete $set_a{$_}, delete $set_b{$_}
}
elsif ( $res < 0 ) {
$set_b{$_} = delete($set_a{$_});
}
else {
$set_a{$_} = delete($set_b{$_});
}
}
next if ! exists $set_a{$_};
my $m = $_;
for my $n ( keys %set_b ) {
my $gcd = gcd($m, $n);
if ($gcd > 1 ) {
++$set_a{$m / $gcd} if $m != $gcd;
++$set_b{$n / $gcd} if $n != $gcd;
! $set_a{$m} && delete $set_a{$m};
! $set_b{$n} && delete $set_b{$n};
$finished = 0, last;
}
}
}
}
}
__END__
__C__
/* Implementation of Euclid's Algorithm by fizbin */
int gcd(int m, int n) {
while( 1 ) {
if (n==0) {return m;}
m %= n;
if (m==0) {return n;}
n %= m;
}
}
It doesn't produce the output you wanted so I didn't bother trying to optimize it WRT choosing pairings. With the lack of optimizations it is likely not that efficient  but it was fun.
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.

