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 .
Solution 3 (bash)
Let be an even integer. Using the Chicken McNugget Theorem on the two smallest odd composite integers that are relatively prime from each other, 9 and 25, show that the maximum is 191, and the maximum even integer is 190 or lower. We use the fact that sufficiently high multiples of 6, 10, 14, 22, etc. can be represented as . We bash each case until we find one that works.
Solution 4 (easiest)
The easiest method is to notice that any odd number that ends in a 5 is a composite (except for 5 itself). This means that we will have 15, 25, 35, etc... no matter what. What it also means is that if we look at the end digit, if 15 plus another number will equal that number, then any number that has that same end digit can be added by that same number plus a version of 15, 25, 35...
For example, let's say we assume our end digit of the number is 4. If we have 5 as one of our end digits, then 9 must be the end digit of the other number. If we go down our list of numbers that end with a 9 and is composite, we will stumble upon the number 9 itself. That means that the number 15+9 is able to be written in a composite form, but also anything that ends with a 4 and is above 15+9(ex. 34, 44 . . .). Hence the largest number that ends with a 4 that satisfies the conditions is 14.
If you list out all the numbers(15, 27, 9, 21, 33), you will notice that 33 is the largest number where the last digit is not repeated (13 and 23 are not composite). That means that 33+15 and any number greater that ends with a 3 is bad(ex. 58, 68. . .), so the largest number that satisfies the conditions is the largest number that ends with a 8 and is below 48. That number would be
Solution 5
Claim: The answer is .
Proof: It is fairly easy to show 38 can't be split into 2 odd composites.
Assume there exists an even integer that m can't be split into 2 odd composites.
Then, we can consider m modulo 5.
If m = 0 mod 5, we can express m = 15 + 5k for some integer k. Since , so . Thus, 5k is composite. Since 15 is odd and composite, m is even, 5k should be odd as well.
If m = 1 mod 5, we can express m = 21 + 5k for some integer k. Since , so . Thus, 5k is composite. Since 21 is odd and composite, m is even, 5k should be odd as well.
If m = 2 mod 5, we can express m = 27 + 5k for some integer k. Since , so . Thus, 5k is composite. Since 27 is odd and composite, m is even, 5k should be odd as well.
If m = 3 mod 5, we can express m = 33 + 5k for some integer k. Since , so . Thus, 5k is composite. Since 33 is odd and composite, m is even, 5k should be odd as well.
If m = 4 mod 5, we can express m = 9 + 5k for some integer k. Since , so . Thus, 5k is composite. Since 9 is odd and composite, m is even, 5k should be odd as well.
Thus, in all cases we can split m into 2 odd composites, and we get at a contradiction.
-Alexlikemath
Solution 6
All numbers that could possibly work must be where is prime. As previous solutions stated, the maximum number that could possibly work by Chicken McNugget is . We then bash from top to bottom:
1. - refuted
2. - refuted
3. - refuted
4. - refuted
5. - refuted
6. - refuted
7. - refuted
8. - refuted
9. - refuted
10. - refuted
11. - refuted
12. - refuted
13. - refuted
14. - refuted
15. - refuted
16. - refuted
17. - it works!
Because we did a very systematic bash as shown, we are confident the answer is
~Arcticturn
Solution 7 (casework)
As stated above, all numbers that could possibly work must be where is prime. If > 30, we consider by modulo 30. could be 1,7,11,13,17,19,23,29 modulo 30. can be expressed as (+)+(-) for some positive, even less then .
If = , p±4 would both be composite
If = , p±2 would both be composite
If = , p±14 would both be composite
If = , p±8 would both be composite
If = , p±8 would both be composite
If = , p±14 would both be composite
If = , p±2 would both be composite
If = , p±4 would both be composite
So < 30
From here, just try all possible p and find the answer is
~Mathophobia
Video Solution using Bashing
https://www.youtube.com/watch?v=n98zEG1-Hrs ~North America Math Contest Go Go Go
See also
1984 AIME (Problems • Answer Key • Resources) | ||
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Followed by Problem 15 | |
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