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 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. | + | 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 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 <math>38</math>. |
== See also == | == See also == | ||
Revision as of 21:21, 10 February 2009
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 small. Also, we find that is not divisible by 3, 5, 7, and so on. Clearly, must be a prime. We can just check small primes and guess that 
gives us our maximum value of 
.
See also
| 1984 AIME (Problems • Answer Key • Resources) | ||
| 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 | ||
