Cleaning Intel Heatsink (Black Residue & Foil, Replacing CPU)

My husband and I are replacing a motherboard and cpu in an older computer. The cpu is a p4 socket 478 I believe. My question involves removing thermal paste/tape from the heatsink. We are using our original heatsink, and when we removed it, there was a black square on the metal that was scrapable with a fingernail. We figured it was old thermal compound so we removed it. What is left is a silver sticker of some kind.

Do we remove everything down to the bare metal, or is the silver sqaure sticker some kind of pad that should stay there?

Probably a really dumb question, we're inexperienced. Can't seem to find an exact answer anywhere, would appreciate the help.

Thanks,
Gina

Remove anything stuck to the surface of the cpu so the new thermal grease can do it's job. All the cpu info is stamped on the lid, so you don't need any old stickers.