Difference between revisions of "1984 AIME Problems/Problem 14"
(Chicken McNugget Theorem only works if the two numbers in question are relatively prime, so the theorem doesn't work for the pair 9 and 15.) |
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== Solution 3 (bash) == | == Solution 3 (bash) == | ||
| − | Let <math>2n</math> be an even integer. Using the [[Chicken McNugget Theorem]] on the two smallest odd composite integers, 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 <math>n+n</math>. We bash each case until we find one that works. | + | Let <math>2n</math> 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 <math>n+n</math>. We bash each case until we find one that works. |
| − | + | ||
== See also == | == See also == | ||
{{AIME box|year=1984|num-b=13|num-a=15}} | {{AIME box|year=1984|num-b=13|num-a=15}} | ||
Revision as of 07:10, 2 September 2018
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.
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 | ||
