1984 AIME Problems/Problem 14
Contents
Problem
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
Solution 1
Take an even positive integer . is either , , or . Notice that the numbers , , , ... , and in general for nonnegative are odd composites. We now have 3 cases:
If and is , can be expressed as for some nonnegative . Note that and are both odd composites.
If and is , can be expressed as for some nonnegative . Note that and are both odd composites.
If and is , can be expressed as for some nonnegative . Note that and are both odd composites.
Clearly, if , it can be expressed as a sum of 2 odd composites. However, if , it can also be expressed using case 1, and if , using case 3. is the largest even integer that our cases do not cover. If we examine the possible ways of splitting into two addends, we see that no pair of odd composites add to . Therefore, is the largest possible number that is not expressible as the sum of two odd composite numbers.
Solution 2
Let be an integer that cannot be written as the sum of two odd composite numbers. If , then and must all be prime (or , which yields which does not work). Thus and form a prime quintuplet. However, only one prime quintuplet exists as exactly one of those 5 numbers must be divisible by 5.This prime quintuplet is and , yielding a maximal answer of 38. Since , which is prime, the answer is .
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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