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15 September, 04:12

The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$. Find the remainder when $F_{1999}$ is divided by 5.

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  1. 15 September, 04:40
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    The remainder is 1

    Step-by-step explanation:

    Given the Fibonacci sequence

    F_1 = F_2 = 1, and

    F_ (n + 2) = F_ (n + 1) + F_n

    We want to find the remainder when F_ (1999) is divided by 5.

    Let us write the first 20 numbers of the sequence in (mod 5). They are

    F_1 = 1,

    F_2 = 1,

    F_3 = 2,

    F_4 = 3,

    F_5 = 5 = 0 (mod 5),

    F_6 = 3,

    F_7 = 3,

    F_8 = 1

    F_ (9) = 4

    F_ (10) = 0

    F_ (11) = 4

    F_ (12) = 4

    F_ (13) = 3

    F_ (14) = 2

    F_ (15) = 0

    F_ (16) = 2

    F_ (17) = 2

    F_ (18) = 4

    F_ (19) = 1

    F_ (20) = 0

    We have: 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0

    Now, 1999 = 19 (mod 20)

    The 19th number in the sequence is 1.

    So, the remainder is 1.
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