In a certain roleplaying game, the hero is tasked with invading a village of goblins, to slay as many goblins as they can. Each goblin has a health total called hit points (HP), which is a positive integer value. Goblins can have HP as low as 2, for the lowly salt goblins, up to 111, for the mighty goblin lords. A goblin can have any amount of HP between 2 and 111, inclusive. For the following parts, a complete proof is necessary. (a) Goblins are considered to be in the same family if they have the same amount of HP, and a family of goblins is any collection of goblins that are all in the same family. Thinking they are very dark and edgy, the hero slays 666 goblins. Prove that the hero has slain at least 7 goblins that were in the same family. (b) Goblins are considered to be friends with each other if their HP is not coprime (i.e. their HP values share a divisor or factor that is greater than 1). For example, a goblin with 15 HP is friends with