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Four Prime Natural Numbers

XAT 2014 · Quantitative Ability & Data Interpretation · Medium · Number System

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Consider four natural numbers: x,y,x+yx, y, x + y, and x−yx - 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?

  1. A.

    Statement I.

  2. B.

    Statement II.

  3. C.

    Statement I and Statement II.

  4. D.

    Neither Statement I nor Statement II.

  5. E.

    Either Statement I or Statement II.

Answer

A

Explanation

For x,y,x+y,x−yx, y, x+y, x-y to all be prime numbers, one of xx or yy must be 2. If x=2x=2, x−yx-y is not positive prime. Thus y=2y=2. Then x−2,x,x+2x-2, x, x+2 are three consecutive odd primes, which must be 3, 5, 7   ⟹  x=5\implies x=5. The unique set is 5,2,7,35, 2, 7, 3. Their sum =17= 17. Statement I alone is sufficient.

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