The four numbers are (2*3*7)^98, (2*5*7)^70, (3*5*7)^42, and (2*7*11)^28.

Part (III) --

Any number which divides the first three must divide 7^42. There are 43 of these.

Any number which divides the last three must divide 7^28. This is a subset of the 43 already counted.

Any number which divides the 1st, 3rd and 4th must divide 7^28. This is a subset of the 43 already counted.

Finally, any number which divides the 1st, 2nd and 4th must divide (2*7)^28. The first 43 includes all of those which are not a power of 2, so we have 28 powers of 2 * 29 powers of 7 = 812 numbers not included in the 43.

Total which divide at least 3 are 43 + 812 = 855.