AMC 10 Daily Practice Round 4

Complete problem set with solutions and individual problem pages

Problem 12 Easy

Consider the Fibonacci Sequence 1, 1, 2, 3, 5, 8, \cdots. Starting the third number, each succeeding number is the sum of the previous two numbers. Among the first 2023 numbers in the sequence, how many of them are multiples of 5?

  • A.

    403

  • B.

    404

  • C.

    405

  • D.

    505

  • E.

    506

Answer:B

We can determine the pattern of remainders when each term in the sequence is divided by 5: 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, \cdots. We find that a remainder of 0 appears once for every 5 numbers if the divisor is 5. As 2023\div 5=404 \ R \ 3, there are 404 such numbers.

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