The Vault: Robbers

Here I present one of my favourites; a problem of rare invention. From the 1998 Tournament of the Towns:

(Two-person case) Two robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the two robbers (that is, they both get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is even.

(General case) A gang of robbers stole a bag of coins from a merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies).

Prove that after one coin is removed, the number of coins remaining is divisible by the number of robbers.

Solution