19. Little Red Riding Hood is delivering waffles to three grannies. She starts with a basket full of waffles. Just before she enters the house of each granny, the Big Bad Wolf eats half of the waffles in her basket. She delivers the same number of waffles to each granny. When she leaves the third granny's house, she has no waffles left. Which of the following numbers definitely divides the number of waffles she started with?
A 4
B 5
C 6
D 7
E 9
19. D Let $x$ be the number of waffles Little Red Riding Hood delivered to each of the grannies. She delivered $x$ waffles to the third granny and so, since the Big Bad Wolf eats half of the waffles in her basket just before she enters each granny's house, she must have arrived with $2 x$ waffles. Therefore she had $3 x$ waffles before giving the second granny her waffles and hence had $2 \times 3 x=6 x$ waffles when she arrived. Therefore she had $6 x+x=7 x$ waffles before giving the first granny her waffles and hence had $2 \times 7 x=14 x$ waffles in her basket when she arrived. Hence, since we do not know the value of $x$, the only numbers that definitely divide the number of waffles she started with are the numbers that divide 14 , namely $1,2,7$ and 14. Therefore, of the options given, only 7 definitely divides the number of waffles she started with.