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Math Help - Why is 5 the answer to this question?

  1. #1
    Junior Member NowIsForever's Avatar
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    Why is 5 the answer to this question?

    Looking through unanswered posts on Yahoo! Answers, while trying to find one that had compelling interest, I found this one:

    Find the digit that appears immediately before the decimal point in: (8 + sq root(59))^(2013!)?

    Was trying to figure this out the whole night. Please help!

    • 4 days ago
    • - 12 hours left to answer.


    Find the digit that appears immediately before the decimal point in: (8 + sq root(59))^(2013!)? - Yahoo! Answers


    Like the OP I have been scratching my head on this one. Can anybody help?
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  2. #2
    Junior Member NowIsForever's Avatar
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    Re: Why is 5 the answer to this question?

    I posted the same question in the Drexel Math Forum, and got this result (reply to that post) by Ara M Jamboulian:

    Ara's Idea expanded goes like this:

    (1) Let a = 8 + sqrt(59) and b = 8 - sqrt(59)

    (2) Prove by induction that a^k + b^k is congruent to 6 mod 10 for any integer k >= 2.

       Defining c_k and d_k in this way

        a^k=c_k+d_k\sqrt{59}\hspace{8}and\hspace{8}b^k=c_k-d_k\sqrt{59}

       We see that

        c_{k+1}=8c_k+59d_k\hspace{8}and\hspace{8}d_{k+1}=8  d_k+c_k

       and this can be demonstrated via induction.

       Let the ordered pair of digits (c,d)_k represent the condition that c_k\hspace{4}mod\hspace{4}10\equiv c and d_k\hspace{4}mod\hspace{4}10 \equiv d

       Then

        (8,1)_1 \rightarrow (3,6)_2 \rightarrow (8,1)_3, etc.

       Since

        \hspace{8}a^k+b^k=2c_k

       It follows that

        \hspace{8}a^k+b^k\equiv\6\hspace{4}mod\hspace{4}10  \hspace{8}(8\rightarrow 6\hspace{8}and\hspace{8}3\rightarrow 6)


    (3) Also prove that floor[a^k]=a^k + b^k - 1

       This follows easily since b^k being perpetually less than 1 implies that floor[a^k] = 2c_k -1 = a^k+b^k - 1.

    (4) It follows that floor[a^k] is congruent to 5 mod 10.

    (Thanks to both the OP and Ara for this interesting problem and help towards its solution.)


    I wondered if the numbers 8 and 59 had any special significance. There seems to be none. Some numbers cause the last digit to oscillate through a series of digits. Perhaps there are some surprises in the more general situation, but intuitively I think not.
    Last edited by NowIsForever; April 24th 2013 at 04:57 AM.
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