2009 AMC 10A Problems/Problem 19
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 cirle 's trip. How many possible values can have?
Solution
The circumference of circle A is , and the circumference of circle B with radius is . Since circle B makes a complete revolution and ends up on the same point, the circumference of A must be a perfect factor of the circumference of B, therefore the quotient must be an integer.
R must then be a factor of 100, excluding 100 (because then circle B would be the same size as circle A). . 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 | |
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