Difference between revisions of "2016 AMC 12A Problems/Problem 5"

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==Solution==
 
==Solution==
In this case, a counterexample is a number that would prove Goldbach's conjecture false. Therefore, the answer is
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In this case, a counterexample is a number that would prove Goldbach's conjecture false. The conjecture asserts what '''can''' be done with '''even''' integers greater than 2.
Therefore the solution is <cmath>{\textbf{(D)}\text{ an even integer greater than 2 that cannot be written as the sum of two prime numbers.}}</cmath>
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Therefore the solution is <cmath>{\textbf{(E)}\text{ an even integer greater than 2 that cannot be written as the sum of two prime numbers.}}</cmath>
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=A|num-b=3|num-a=5}}
 
{{AMC12 box|year=2016|ab=A|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 05:11, 4 February 2016

Problem

Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?

$\textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}\\ \qquad\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}\\ \qquad\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}\\ \qquad\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$


Solution

In this case, a counterexample is a number that would prove Goldbach's conjecture false. The conjecture asserts what can be done with even integers greater than 2. Therefore the solution is \[{\textbf{(E)}\text{ an even integer greater than 2 that cannot be written as the sum of two prime numbers.}}\]

See Also

2016 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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