Difference between revisions of "2019 AMC 10A Problems/Problem 25"
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==Solution== | ==Solution== | ||
− | The main insight is that <cmath>\frac{(n^2)!}{(n!)^{n+1}}</cmath> is always an integer. This is true because it is precisely the number of ways to split up <math>n^2</math> objects into <math>n</math> unordered groups of size <math>n</math>, which has to be an integer. Thus, <cmath>\frac{(n^2-1)!}{(n!)^n}</cmath> is not an integer | + | The main insight is that <cmath>\frac{(n^2)!}{(n!)^{n+1}}</cmath> is always an integer. This is true because it is precisely the number of ways to split up <math>n^2</math> objects into <math>n</math> unordered groups of size <math>n</math>, which has to be an integer. Thus, <cmath>\frac{(n^2-1)!}{(n!)^n}</cmath> is not an integer if <math>n^2 \nmid n!</math>. This happens only for <math>n=4</math> or <math>n</math> prime. Thus, there are <math>16</math> integers that do not work, which means the answer is <math>\boxed{\mathbf{(D)}\ 34}</math>. |
==See Also== | ==See Also== |
Revision as of 22:17, 9 February 2019
- The following problem is from both the 2019 AMC 10A #25 and 2019 AMC 12A #24, so both problems redirect to this page.
Problem
For how many integers between and , inclusive, is an integer? (Recall that .)
Solution
The main insight is that is always an integer. This is true because it is precisely the number of ways to split up objects into unordered groups of size , which has to be an integer. Thus, is not an integer if . This happens only for or prime. Thus, there are integers that do not work, which means the answer is .
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 |
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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.