Difference between revisions of "1984 AIME Problems/Problem 14"
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| − | Let <math>n</math> be an integer that cannot be written as the sum of two odd composite numbers. If <math>n>33</math>, then <math>n-9,n-15,n-21,n-25,n-27,</math> and <math>n-33</math> must all be prime (or <math>n-33=1</math>, which yields <math>n=34=9+25</math> which does not work). Thus <math>n-9,n-15,n-21,n-27,</math> and <math>n-33</math> 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 <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answer is <math>\boxed{038}</math>. | + | Let <math>n</math> be an integer that cannot be written as the sum of two odd composite numbers. If <math>n>33</math>, then <math>n-9,n-15,n-21,n-25,n-27,</math> and <math>n-33</math> must all be prime (or <math>n-33=1</math>, which yields <math>n=34=9+25</math> which does not work). Thus <math>n-9,n-15,n-21,n-27,</math> and <math>n-33</math> form a sexy prime quintuplet. However, only one sexy prime quintuplet exists as exactly one of those 5 numbers must be divisible by 5.This sexy prime quintuplet is <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answer is <math>\boxed{038}</math>. |
== See also == | == See also == | ||
Revision as of 17:49, 24 May 2016
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 sexy prime quintuplet. However, only one sexy prime quintuplet exists as exactly one of those 5 numbers must be divisible by 5.This sexy 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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
