Sample Question 5:

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?

(A) an odd integer greater than 2 that can be written as the sum of two prime numbers

(B) an odd integer greater than 2 that cannot be written as the sum of two prime numbers

(C) an even integer greater than 2 that can be written as the sum of two numbers that are not prime

(D) an even integer greater than 2 that can be written as the sum of two prime numbers

(E) an even integer greater than 2 that cannot be written as the sum of two prime numbers

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Answer Keys

Question 5: E

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Solutions

Question 5
Step 1: Understand that a counterexample here would be a scenario that contradicts Goldbach's conjecture.
Step 2: Recall that Goldbach's conjecture says every even integer greater than 2 can be written as the sum of two prime numbers.
Step 3: Notice that the conjecture is specifically about even integers greater than 2 and their relation to prime numbers.
Step 4: Determine that anything contradicting this would be an even integer greater than 2, that cannot be expressed as the sum of two prime numbers.
Step 5: Finally, we identify that answer choice $\textbf{(E)}$ describes a scenario that would contradict Goldbach's conjecture: an even integer greater than 2 that cannot be written as the sum of two prime numbers. Thus, $\textbf{(E)}$ is the correct answer.