AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 20 Medium

Represent 77 as the sum of n consecutive positive integers, where n \geq 2. How many possible values of n are there?

  • A.

    4

  • B.

    3

  • C.

    2

  • D.

    1

  • E.

    0

Answer:B

77=\frac{11\times 7\times 2}{2}.

77=\frac{(11\times 7)\times 2}{2}=\frac{(38+39)\times 2}{2}=38+39;

77=\frac{(11\times 2)\times 7}{2}=\frac{(8+14)\times 7}{2}=8+9+10+11+12+13+14;

77=\frac{(2\times 7)\times 11}{2}=\frac{(2+12)\times 11}{2}=2+3+4+5+6+7+8+9+10+11+12.

Thus, there are 3 possible values of n.

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