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Problem
Circle has radius . Circle has an integer radius and remains internally tangent to circle as it rolls once around the circumference of circle . The two circles have the same points of tangency at the beginning and end of circle 's trip. How many possible values can have?
Solution
The circumference of circle is , and the circumference of circle with radius is . Since circle makes a complete revolution and ends up on the same point, the circumference of must be a multiple of the circumference of , therefore the quotient must be an integer.
Thus, .
Therefore must then be a factor of , excluding because the problem says that . . Therefore 100 has factors*. But you need to subtract 1 from 9, in order to exclude 100. Therefore the answer is .
*The number of factors of and so on, where and are prime numbers, is .
See Also
2009 AMC 10A (Problems • Answer Key • Resources)  
Preceded by Problem 18 
Followed by Problem 20  
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25  
All AMC 10 Problems and Solutions 
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