Difference between revisions of "1984 AIME Problems/Problem 14"
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== Solution == | == Solution == | ||
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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. 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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== See also == | == See also == | ||
* [[1984 AIME Problems/Problem 13 | Previous problem]] | * [[1984 AIME Problems/Problem 13 | Previous problem]] | ||
* [[1984 AIME Problems/Problem 15 | Next problem]] | * [[1984 AIME Problems/Problem 15 | Next problem]] | ||
* [[1984 AIME Problems]] | * [[1984 AIME Problems]] | ||
Revision as of 17:31, 6 March 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 
for some positive integer 
. Notice that we must have 
, 
, 
, 
, ..., 
all prime for every odd composite number 
less than . Therefore must be pretty small. Also, we find that is not divisible by 3, 5, 7, and so on. Clearly, must be a prime. Um, we can just check small primes and guess that 
gives us our maximum value of 38.
