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Math Help - Proving gcd(5^98 + 3, 5^99 +1 ) = 14

  1. #1
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    Proving gcd(5^98 + 3, 5^99 +1 ) = 14

    Hello,

    this is my first time using a forum for mathematics.

    I have come across a question that I am unable to solve.

    Prove that gcd(5^98 + 3, 5^99 +1) = 14

    I've been thinking about it and I believe that using the euclidean algorithm would work, but I am unsure on how to approach it.
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  2. #2
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    TheEmptySet's Avatar
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    Your idea is sound. Here is a hint note that

    Not that 5^{98}+3 is even so 2 divides it and

    by fermat's little theorem 5^{6} \equiv 1 \test{mod}(7)

    Use these two facts to show that 14 divides 5^{98}+3

    and so the euclidean algorithm terminates.
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