Difference between revisions of "2002 AMC 12P Problems/Problem 12"
The 76923th (talk | contribs) m (→Solution) |
The 76923th (talk | contribs) m (→Solution) |
||
| Line 17: | Line 17: | ||
Since this is a number theory question, it is clear that the main challenge here is factoring the given cubic. In general, the rational root theorem will be very useful for these situations. | Since this is a number theory question, it is clear that the main challenge here is factoring the given cubic. In general, the rational root theorem will be very useful for these situations. | ||
| − | The rational root theorem states that all rational roots of <math>n^3 - 8n^2 + 20n - 13</math> will be among <math>1, 13, -1, -13</math>. | + | The rational root theorem states that all rational roots of <math>n^3 - 8n^2 + 20n - 13</math> will be among <math>1, 13, -1</math>, and <math>-13</math>. |
== See also == | == See also == | ||
{{AMC12 box|year=2002|ab=P|num-b=11|num-a=13}} | {{AMC12 box|year=2002|ab=P|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 15:01, 10 March 2024
Problem
For how many positive integers 
is 
a prime number?
Solution
Since this is a number theory question, it is clear that the main challenge here is factoring the given cubic. In general, the rational root theorem will be very useful for these situations.
The rational root theorem states that all rational roots of will be among 
, and 
.
See also
| 2002 AMC 12P (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
