Let S be a subset of {1, 2, 3, 4...., 100}, containing 50 elements. How many such sets have the property that every pair of numbers in S has a common divisor that is greater than 1?

A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one 1x1 square, two 2x2, two 3x3, and the one 4x4