Difference between revisions of "2015 AIME I Problems/Problem 3"
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==Solution== | ==Solution== | ||
− | + | Let the positive integer mentioned be <math>a</math>, so that <math>a^3 = 16p+1</math>. Note that <math>a</math> must be odd, because <math>16p+1</math> is odd. | |
− | <math>a^3 = | + | Rearrange this expression and factor the left side (this factoring can be done using <math>(a^3-b^3) = (a-b)(a^2+a b+b^2)</math>, or synthetic divison once it is realized that <math>a = 1</math> is a root): |
<math>a^3-1 = 16p</math> | <math>a^3-1 = 16p</math> | ||
− | + | <math>(a-1)(a^2+a+1) = 16p</math> | |
− | <math> | + | Because <math>a</math> is odd, <math>a-1</math> is even and <math>a^2+a+1</math> is odd. If <math>a^2+a+1</math> is odd, <math>a-1</math> must be some multiple of <math>16</math>. However, for <math>a-1</math> to be any multiple of <math>16</math> other than <math>16</math> would mean <math>p</math> is not a prime. Therefore, <math>a-1 = 16</math> and <math>a = 17</math>. |
− | + | Then our other factor, <math>a^2+a+1</math>, is the prime <math>p</math>: | |
− | + | <math>(a-1)(a^2+a+1) = 16p</math> | |
− | + | <math>(17-1)(17^2+17+1) =16p</math> | |
+ | |||
+ | <math>p = 289+17+1 = \boxed{307}</math>. | ||
== See also == | == See also == |
Revision as of 15:14, 20 March 2015
Problem
There is a prime number such that is the cube of a positive integer. Find .
Solution
Let the positive integer mentioned be , so that . Note that must be odd, because is odd.
Rearrange this expression and factor the left side (this factoring can be done using , or synthetic divison once it is realized that is a root):
Because is odd, is even and is odd. If is odd, must be some multiple of . However, for to be any multiple of other than would mean is not a prime. Therefore, and .
Then our other factor, , is the prime :
.
See also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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