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 circle
'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 multiple of the circumference of B, therefore the quotient must be an integer.
Thus,
Therefore 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 ofand 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.