Remote central locking doesn't work, but boot release does!
 

I went to view a 2001 A3 1.6 Sport yesterday evening. It was in immaculate condition, however there is one problem. The central locking doesn't lock ro unlock the doors, it makes the hazards flash and the interior alarm led flash too, but the doors just don't lock. The odd thing is that the boot release does work! I am able to lock the doors manual, but even then the 'central locking' doesn't work and both doors have to be locked individually. The car had been stood for a while so it may just be a weak battery as i had a similar problem in a previous car. However i am concerned as i have read stories about the locking pump in the boot breaking and it being an expensive job to repair. I go to test drive tonight so will be able to determine whether or not it is the battery. Is there any way to determine the cause of the problem. I had thought about reprogramming, but wouldn't an 'un-programmed' key not work at all (i.e. not make the lights flash)? Plus it only has the one key so to reprogramme i would need to have another cut.
Thanks :)