Difference between revisions of "2009 AMC 10A Problems/Problem 19"
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*The number of factors of <math>a^x\: \cdot \: b^y\: \cdot \: c^z\;...</math> and so on, where <math>a, b,</math> and <math>c</math> are prime numbers, is <math>(x+1)(y+1)(z+1)...</math>. | *The number of factors of <math>a^x\: \cdot \: b^y\: \cdot \: c^z\;...</math> and so on, where <math>a, b,</math> and <math>c</math> are prime numbers, is <math>(x+1)(y+1)(z+1)...</math>. | ||
| − | *??? | + | *??? |
== See Also == | == See Also == | ||
Revision as of 22:28, 3 December 2019
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 ofand so on, where
and
are prime numbers, is
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
