Difference between revisions of "2009 AMC 10A Problems/Problem 19"
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*The number of factors of <math>a^x\: *\: b^y\: *\: c^z\;...</math> and so on, is <math>(x+1)(y+1)(z+1)...</math>. | *The number of factors of <math>a^x\: *\: b^y\: *\: c^z\;...</math> and so on, is <math>(x+1)(y+1)(z+1)...</math>. | ||
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Revision as of 18:57, 18 February 2009
The circumference of circle A is 200
, 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, is 
.
and so on, is 
.| 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 | ||
