Dorothy organized a party for the birthday of Duck Mom and she also prepared a cylindershaped cake. Since she was originally expecting to have $15$ guests, she divided the top of the cake into this many equal circular sectors, marking where the cuts need to be made. Just for fun Dorothy’s brother Donald split the top of the cake into $10$ equal circular sectors in such a way that some of the radii that he marked coincided with Dorothy’s original markings. Just before the arrival of the guests Douglas cut the cake according to all markings, and then he placed the cake into the fridge. This way they forgot about the cake and only got to eating it when only $6$ of them remained. Is it possible for them to divide the cake into $6$ equal parts without making any further cuts?