Difference between revisions of "1984 AIME Problems/Problem 14"

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== Solution ==
 
== Solution ==
  
Let the desired integer be <math>2n</math> for some positive integer <math>n</math>. Notice that we must have <math>2n-9</math>, <math>2n-15</math>, <math>2n-21</math>, <math>2n-25</math>, ..., <math>2n-k</math> all prime for every odd composite number <math>k</math> less than <math>2n</math>. Therefore <math>n</math> must be pretty small. Also, we find that <math>n</math> is not divisible by 3, 5, 7, and so on. Clearly, <math>n</math> must be a prime. Um, we can just check small primes and guess that <math>n=19</math> gives us our maximum value of 38.
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Let the desired integer be <math>2n</math> for some positive integer <math>n</math>. Notice that we must have <math>2n-9</math>, <math>2n-15</math>, <math>2n-21</math>, <math>2n-25</math>, ..., <math>2n-k</math> all prime for every odd composite number <math>k</math> less than <math>2n</math>. Therefore <math>n</math> must be pretty small. Also, we find that <math>n</math> is not divisible by 3, 5, 7, and so on. Clearly, <math>n</math> must be a prime. We can just check small primes and guess that <math>n=19</math> gives us our maximum value of 38.
  
 
== See also ==
 
== See also ==

Revision as of 09:00, 15 October 2007

Problem

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

Solution

Let the desired integer be $2n$ for some positive integer $n$. Notice that we must have $2n-9$, $2n-15$, $2n-21$, $2n-25$, ..., $2n-k$ all prime for every odd composite number $k$ less than $2n$. Therefore $n$ must be pretty small. Also, we find that $n$ is not divisible by 3, 5, 7, and so on. Clearly, $n$ must be a prime. We can just check small primes and guess that $n=19$ gives us our maximum value of 38.

See also

1984 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions