Heres my go without reviewing your code first.

use strict;
use warnings;
use overload '""'=>'stringify';
sub stringify { $_[0]->() };

sub make_sub {
    my ($N) = ( my @f = sort {$a <=> $b} @_ );
    return bless sub{ for(;;$N++){ $N % $_ || return $N++ for @f }  };
}

@ARGV=(2,3,5) unless @ARGV;
@ARGV=sort @ARGV;

{
    local $"=", ";
    local $\="\n";
    print "Factors: @ARGV";

    foreach  (@ARGV) {
        my $f=make_sub($_);
        print "F($_) = @{[($f) x 25]}";
    }
    my $o=make_sub(@ARGV);
    print "R(@ARGV) = @{[($o) x 25]}";
    print "----";
}

__END__

Factors: 2, 3, 5
F(2) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34,
+ 36, 38, 40, 42, 44, 46, 48, 50
F(3) = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51
+, 54, 57, 60, 63, 66, 69, 72, 75
F(5) = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 
+85, 90, 95, 100, 105, 110, 115, 120, 125
R(2, 3, 5) = 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 
+24, 25, 26, 27, 28, 30, 32, 33, 34
----
[download]

Update: It looks like this is the lazy version of the iterative solution by kvale.
Update2: BrowserUk correctly points out the sort needs to be numeric, which ive fixed.

Update3: I tweaked my original a little to support a DESTROY method, and added a second more efficient variant.
Update4: Added readmore around original code.

use strict;
use warnings;
use overload '""'=>'stringify';
sub stringify { $_[0]->() };

sub lcf {
    my @f=sort {$a<=>$b} @_;
    my @v=shift @f;
    for my $f (@f) {
        push @v,$f
            unless grep !($f % $_),@v;
    }
    @v
}

sub make_sub1 {
    my ($N) =  my @f = lcf( @_ );
    return bless sub{
        return "Simple Factors = [ @f ] Cur = $N" if @_;
        for(;;$N++){ $N % $_ || return $N++ for @f }
    };
}


sub make_sub2 {
    my @f = my @v = lcf( @_ );
    return bless sub {
        return "Smart  Factors = [ @f ] Cur = [ @v ]" if @_;
        my ($min,$ret);
        do {
            $min = 0;
            $v[$min] >= $v[$_] and $min=$_ for 1..$#v;
            $ret = $v[$min];
            $v[$min] += $f[$min];
        } while grep !($ret % $f[$_]), 0 .. $min-1;
        return $ret;
    };
}

sub DESTROY {
    my $s=shift;
    warn "DESTROY: $s ",$s->(1),"\n";
}

@ARGV=(2,3,4,5,6) unless @ARGV;
@ARGV=sort @ARGV;

{
    local $"=", ";
    local $\="\n";
    print "Factors: @ARGV";

    foreach  (@ARGV) {
        my $f=make_sub1($_);
        print "F($_) = @{[($f) x 25]}";
    }
    for my $o ( make_sub1(@ARGV),make_sub2(@ARGV)) {
        print "R(@ARGV) = @{[($o) x 25]}";
    }
    print "----";
}
__END__
DESTROY: main=CODE(0x1ac5130) Simple Factors = [ 2 ] Cur = 51
DESTROY: main=CODE(0x1ac5130) Simple Factors = [ 3 ] Cur = 76
DESTROY: main=CODE(0x1ac5130) Simple Factors = [ 4 ] Cur = 101
DESTROY: main=CODE(0x1ac5130) Simple Factors = [ 5 ] Cur = 126
DESTROY: main=CODE(0x1ac5130) Simple Factors = [ 6 ] Cur = 151
DESTROY: main=CODE(0x1ac21c8) Smart  Factors = [ 2, 3, 5 ] Cur = [ 36,
+ 36, 35 ]
DESTROY: main=CODE(0x1c0535c) Simple Factors = [ 2, 3, 5 ] Cur = 35
Factors: 2, 3, 4, 5, 6
F(2) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34,
+ 36, 38, 40, 42, 44, 46, 48, 50
F(3) = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51
+, 54, 57, 60, 63, 66, 69, 72, 75
F(4) = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 6
+8, 72, 76, 80, 84, 88, 92, 96, 100
F(5) = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 
+85, 90, 95, 100, 105, 110, 115, 120, 125
F(6) = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 
+102, 108, 114, 120, 126, 132, 138, 144, 150
R(2, 3, 4, 5, 6) = 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21
+, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34
R(2, 3, 4, 5, 6) = 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21
+, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34
----
[download]