1984 AIME Problems/Problem 14

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 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.

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
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1984 AIME Problems/Problem 14

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