Problem 9 · 2024 Math Kangaroo
Medium
Number Theory
divisibility
50 children are sitting in a circle. They throw a ball. Each child that gets the ball throws it to the child sitting six places to their left. Frida gets the ball 100 times during the game. How many children never get the ball during this time?
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Answer: D — 25
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Hint 1 of 2
The ball lands on positions that are multiples of 6 around a circle of 50.
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Hint 2 of 2
How many distinct seats it visits depends on the greatest common divisor of 6 and 50.
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Approach: count the orbit length using gcd
- Starting from one child, the ball moves 6 seats each throw around 50 seats.
- The seats it reaches are spaced by gcd(6, 50) = 2, so it visits 50 ÷ 2 = 25 children.
- The other 50 − 25 = 25 children never get the ball.
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