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Topic: Why is 5 the answer to this question?
Replies: 2   Last Post: Apr 23, 2013 8:59 PM

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Ara M Jamboulian

Posts: 65
Registered: 12/6/04
Re: Why is 5 the answer to this question?
Posted: Apr 21, 2013 5:34 PM
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>
>
> Find the digit that appears immediately before the
> decimal point in: (8 + sq root(59))^(2013!)?
>
>

Idea:
Let a = 8 + sqrt(59) and b = 8 - sqrt(59)
Prove by induction that a^k + b^k is congruent to 6 mod 10 for any integer k >= 2.
Also prove that floor[a^k]=a^k + b^k - 1
It follows that floor[a^k] is congruent to 5 mod 10



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