19. A group of 50 girls sit in a circle. They throw a ball around the circle. Each girl who gets the ball throws it to the girl sitting six places anticlockwise from her, who catches it. Freda catches the ball 100 times. In that time, how many girls never get to catch the ball?
A 0
B 8
C 10
D 25
E 40
SolUTION
For Freda to catch the ball a second time, the ball must pass a multiple of 50 girls. Since the ball is thrown to the girl sitting six places anticlockwise from the thrower each time, the ball must also pass a multiple of 6 girls. The Lowest Common Multiple of 6 and 50 is 150 and hence the ball will have passed 150 girls before Freda catches the ball for a second time. In that time, the number of girls who have caught the ball is $150 \div 6=25$. This pattern will then continue with the same 25 girls, including Freda, catching the ball before anyone catches it for a further time and so on. Therefore, the number of girls who never get to catch the ball is $50-25=25$.