Rokhlin's theorem

I will sketch a proof of the version of Rokhlin's theorem which states that the signature of every smooth closed spin 4 manifold is divisible by 16, assuming some other big theorems like Hirzebruch signature theorem. In particular, I want to discuss where the mysterious number 16 comes from. Somehow it is related to the third stable homotopy group of spheres, which is Z/24Z.