2016 AMC 12A Problem 5

Question

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?

an odd integer greater than

2

that can be written as the sum of two prime numbers

an odd integer greater than

2

that cannot be written as the sum of two prime numbers

an even integer greater than

2

that can be written as the sum of two numbers that are not prime

an even integer greater than

2

that can be written as the sum of two prime numbers

an even integer greater than

2

that cannot be written as the sum of two prime numbers

Answer: E

Concepts:counterexample logical deduction

Difficulty rating: 1100

Small hint:

A counterexample keeps the hypothesis but breaks the conclusion

Big hint:

The hypothesis is “even integer greater than

2

”; the conclusion is “a sum of two primes”

Solution:

A counterexample must satisfy the hypothesis of being an even integer greater than

2

while failing the conclusion that it can be written as the sum of two prime numbers.

Thus, the correct answer is E.

2016 AMC 12A Problem 5

Answer: E

Difficulty rating: 1100

Small hint:

Big hint:

Solution:

Problem 5 in other years