HELP! Car won't pass smog!

Can someone please help me? I just bought a car from a guy who promised that the car was in good working condition, but hadn't had it smogged. I just took it to get smogged, and it passed the emissions part of the test but it failed the OBD II test. The mechanic told me to drive it around for about a week so that the computer could gain "memory" because he thought that maybe it had been reset. I need to register this car in 6 days, which means I have to have it re-smogged SOON. Can anyone tell me what the problem might be and how I can fix it with little to no cost to me?

Thank you in advance for all your help!

did your mechanic get/give you a diagnostic code? it's a sure-fire way to figure out what's wrong... a few examples i found on the net were; P0440 code would mean you've got a fuel vapor leak. P0300 means a random misfire, P1406 would be your EGR valve. P0131 to P0161 all deal with failure of your oxygen sensor or heater...

sorry if i couldn't be much help