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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.

Cheers - L~R

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