Question

Consider four natural numbers: $x, y, x + y$ , and $x - y$ . Two statements are provided below: I. All four numbers are prime numbers. II. The arithmetic mean of the numbers is greater than 4.

Which of the following statements would be sufficient to determine the sum of the four numbers?

Accepted Answer

A. For $x, y, x+y, x-y$ to all be prime numbers, one of $x$ or $y$ must be 2. If $x=2$, $x-y$ is not positive prime. Thus $y=2$. Then $x-2, x, x+2$ are three consecutive odd primes, which must be 3, 5, 7 $\implies x=5$. The unique set is $5, 2, 7, 3$. Their sum $= 17$. Statement I alone is sufficient.

Four Prime Natural Numbers

Explanation

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