Grades 7-8 Video Solutions 2024-2024_7-8

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Problem number 24. A group of 50 students are sitting in a circle and throwing a ball.  Each student who gets the ball throws it to the 6th student sitting counterclockwise from where they are sitting, who catches it.  Freyja catches the ball 100 times. At that time, how many students never get to catch the ball?  First, we have to notice two things. If a student receives the ball again, like Freyja here, after they've given it up,  that means that the ball must have bypassed a multiple of 50 students because there are 50 students in the circle.  That means that if the ball had passed 50 students, it would get back to you. If it passed 100 students, it would get back to you, and so on.  We also know that the ball passes 6 students at a time, so that means that the ball also bypasses a multiple of 6 students.  That means that for, for example, Freyja or any first student to receive the ball after they've already given it up,  it must bypass the least common multiple of 50 and 6 students, which is 150.  And before that occurs, since it has to pass 150 students, the ball must have been passed to 150 divided by 6 equals 25 students,  since in each step we will bypass 6 students. And we can be sure that no two students here are the same, because that would take at least 150 passes.  So this means that between the time someone gives up the ball and them receiving it, it would have taken 25 students in the middle.  And after they receive the ball again, then basically we're restarting the same cycle, because the first person, whoever started, or in this case Freyja, will have the ball again.  And Freyja will keep passing it to the same group of 25 students before it gets to her, and then it passes to another 25 students, and so on and so forth.  As in this case, the same 25 players will get the ball again and again, being the other 25 will never touch the ball.  And so in this time, the number of students who never get to catch the ball is D, 25.

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