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Difference between revisions of "1984 AIME Problems/Problem 14"

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Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be expressed using case 1, and if <math>x = 40</math>, using case 3. <math>38</math> is the largest even integer that our cases do not cover. If we examine the possible ways of splitting <math>38</math> into two addends, we see that no pair of odd composites add to <math>38</math>. Therefore, <math>38</math> is the largest possible number that is not expressible as the sum of two odd composite numbers.
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Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be expressed using case 1, and if <math>x = 40</math>, using case 3. <math>38</math> is the largest even integer that our cases do not cover. If we examine the possible ways of splitting <math>38</math> into two addends, we see that no pair of odd composites add to <math>38</math>. Therefore, <math>\boxed{038}</math> is the largest possible number that is not expressible as the sum of two odd composite numbers.
  
 
== See also ==
 
== See also ==

Revision as of 01:35, 3 March 2011

Problem

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

Solution

Take an even positive integer $x$. $x$ is either $0 \bmod{6}$, $2 \bmod{6}$, or $4 \bmod{6}$. Notice that the numbers $9$, $15$, $21$, ... , and in general $9 + 6n$ for nonnegative $n$ are odd composites. We now have 3 cases:

If $x \ge 18$ and is $0 \bmod{6}$, $x$ can be expressed as $9 + (9+6n)$ for some nonnegative $n$. Note that $9$ and $9+6n$ are both odd composites.

If $x\ge 44$ and is $2 \bmod{6}$, $x$ can be expressed as $35 + (9+6n)$ for some nonnegative $n$. Note that $35$ and $9+6n$ are both odd composites.

If $x\ge 34$ and is $4 \bmod{6}$, $x$ can be expressed as $25 + (9+6n)$ for some nonnegative $n$. Note that $25$ and $9+6n$ are both odd composites.


Clearly, if $x \ge 44$, it can be expressed as a sum of 2 odd composites. However, if $x = 42$, it can also be expressed using case 1, and if $x = 40$, using case 3. $38$ is the largest even integer that our cases do not cover. If we examine the possible ways of splitting $38$ into two addends, we see that no pair of odd composites add to $38$. Therefore, $\boxed{038}$ is the largest possible number that is not expressible as the sum of two odd composite numbers.

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
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