Blank Macbook pro with no operating system. Trying to fix

Question

I'm trying to fix a 2010 MacBook Pro. It had Yosemite 10.10. It was running slow, so I performed a factory reset. Unfortunately, when I tried to re-install Yosemite 10.10, it prompted me with an Apple sign in. I used my mum's Apple ID (it's her MacBook), and that didn't work. So I'm trying to find a way to install an earlier version of mac os, but I have to try and do it in a way where I don't have to use an email address (We don't know the original Apple ID used with this laptop).