2022 AMC 8

Complete problem set with solutions and individual problem pages

Problem 13 Medium

How many positive integers can fill the blank in the sentence below?

"One positive integer is            more than twice another, and the sum of the two numbers is 28."

  • A.

    6

  • B.

    7

  • C.

    8

  • D.

    9

  • E.

    10

Answer:D

Let m and n be positive integers such that m>n and m+n=28. It follows that m=2n+d for some positive integer d. We wish to find the number of possible values for d.

By substitution, we have (2n+d)+n=28, from which d=28-3n. Note that n=1,2,3,\ldots,9 each generate a positive integer for d, so there are \boxed{\textbf{(D) } 9} possible values for d.

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