Difference between revisions of "2005 AMC 10B Problems/Problem 22"
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For how many positive integers <math>n</math> less than or equal to <math>24</math> is <math>n!</math> evenly divisible by <math>1 + 2 + \ldots + n</math>? | For how many positive integers <math>n</math> less than or equal to <math>24</math> is <math>n!</math> evenly divisible by <math>1 + 2 + \ldots + n</math>? | ||
| − | <math>\text{(A) 162} \text{(B) 180} \text{(C) 324} \text{(D) 360} \text{(E) 720}</math> | + | <math>\text{(A) 162 } \text{(B) 180 } \text{(C) 324 } \text{(D) 360 } \text{(E) 720 }</math> |
== Solution == | == Solution == | ||
Revision as of 23:13, 23 January 2015
Problem
For how many positive integers 
less than or equal to 
is 
evenly divisible by 
?
Solution
Since 
, the condition is equivalent to having an integer value for 
. This reduces, when 
, to having an integer value for 
. This fraction is an integer unless 
is an odd prime. There are 8 odd primes less than or equal to 25, so there are 
numbers less than or equal to 24 that satisfy the condition.
See Also
| 2005 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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. 
