pull-ups

I was wondering about pull-ups. How many pull-ups can you do? Be honest. The other day my friend challenged me. I was able to do 13 of them and he was able to do 11. I'm talking about pull-ups with good form, not half-assed and cheating.

Can you suggest me a strategy to building up my core. Example I can bench 250 lb no problem but I can't do very many sit-ups or pull-ups. I think that my core is very weak even though I might seem like I'm strong. also do you guys ever train your core?