How to find out this girls last name without asking her?

[Jeremy]

There are different things you can do. Assuming you are old enough to drive or have any form of picture ID. Claim that your school ID/State ID/Driver's License is the WORST photo ever. It doesn't really have to be the worst ever, just pretend like it is. Show it to her. Then play it up like I bet mine is worse than yours. She will invariably show her yours...you have her last name.

Edit--Nearly every school I went to had a photo ID they assigned their students, even junior high and high school. Okay, if you don't have any kind of picture ID. Try looking her up online. Does she use social media? Do you have her phone number? You could always run a reverse phone number lookup on a site like 1800ussearch or intelius to get the person paying the bill for the phone. That may be her parents, or might not. I'd try social media first though, that's free.