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Math Help - Find gcd of 7^(4n+3) − 4^(10n+3) + 11^(10n+2)

  1. #1
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    Find gcd of 7^(4n+3) − 4^(10n+3) + 11^(10n+2)

    Find gcd of 7^{4n+3}-4^{10n+3}+11^{10n+2}, n \in N_0.

    Any idea?
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  2. #2
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    ... the GCD of that expression and which other one?
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  3. #3
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    Quote Originally Posted by coquitao View Post
    ... the GCD of that expression and which other one?
    I meant GCD of the results of this expresion for all n.
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  4. #4
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    I gooot you... Then, the idea is the following: prove that your expression is always divisible by 400 = 16 x 25. Since 400 belongs to the range of it, you are done.
    Last edited by coquitao; September 15th 2009 at 02:51 AM.
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  5. #5
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    Um, just wondering how you got 400 ... cuz I got it by setting n = 0, but when n = 1, it's not divisible by 400 ... but it is divisible by 200 ...
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  6. #6
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    Of course ... I proved the expression always to be divisible by 8 not 16. You are right, Bingk.
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  7. #7
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    You're welcome ... but really, I just did trial and error ... how did you prove that it's always divisible by 8?
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  8. #8
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    The second term is always divisible by 8, so the problem reduces to investigate whether 7^{4n+3}+11^{10n+2} is divisible by 8 or not.

    Since 7^{2} is 1 modulo 8 we have that 7^{4n} is also 1 modulo 8. Therefore 7^{4n+3} \equiv 7 \equiv -1 \mod 8.

    On the other hand, the fact that 11^{2} \equiv 1 \mod 8 implies at once that 11^{10n+2} \equiv 1 \mod 8. Hence 7^{4n+3}+11^{10n+2} \equiv (-1)+1 = 0 \mod 8 as was to be shown.
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