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Math Help - Simple Proof by Induction

  1. #1
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    Simple Proof by Induction

    Using mathematical induction I want to show that 3^{6n}-2^{6n} is divisible by 35,  \forall n \in \mathbb{N}.

    The base case n=1 is true: 3^6-2^6=665, 665/35=19.

    Inductive step: Suppose 3^{6k}-2^{6k} is divisible by 35 for some k \in \mathbb{N}. That means 3^{6k}-2^{6k} = 35m for some m. Then

    3^{6(k+1)}-2^{6(k+1)}= 3^{6k}.3^6-2^{6k}.2^6

    Now how can I simplify this to factor out 3^{6k}-2^{6k} to show that it's divisible by 35 for k+1?
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  2. #2
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    Quote Originally Posted by demode View Post
    3^{6(k+1)}-2^{6(k+1)}= 3^{6k}.3^6-2^{6k}.2^6
    Rewrite 2^6 as 3^6-665.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Without using induction:


    3^{6(k+1)}-2^{6(k+1)}=(3^6)^{k+1}-(2^6)^{k+1}=

    (3^6-2^6)[(3^6)^k+(3^6)^{k-1}2^6+\ldots+(2^6)^k]=35h\;(h\in \mathbb{N})
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