Select any pdf file on your iMac and Command + I (get info). Then choose Open With and select Preview from the drop down menu and close window. Then try opening a pdf. If that works, you don't need Adobe.
Is it only ones you have downloaded? What about for example the ones that came with your iMac - you didn't give a model but mine has a folder "User Guides and Information" that has a number of pdfs. Spotlight should find it for you.