AMC12 2016a Test Paper

Complete the quiz in 75 minutes. Do the quiz as if you are taking the real test. You score will be compared with other students taking the same test to give you a ranking among your peers.
To get a human or AI tutor to help you, click Register

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

(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

Answer Keys

Question 5: E

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.