2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 13 Easy

The positive difference between a pair of primes is equal to 2, and the positive difference between the cubes of the two primes is 31106. What is the sum of the digits of the least prime that is greater than those two primes?

  • A.

    8

  • B.

    10

  • C.

    11

  • D.

    13

  • E.

    16

Answer:E

Let the numbers be a and b, respectively. Then we have a-b=2 and a^3-b^3=31106. With manipulation, (a-b)^2=a^2-2ab+b^2=4 and a^2+ab+b^2=15553, so 3ab=15549 and ab=5183. Observe that a=73 and b=71. The next prime number is 79, so the answer is 7+9=16.

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