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Math Help - Proof using Eulers Theorem

  1. #1
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    Proof using Eulers Theorem

    Prove that (2^15)-(2^3) divides (a^15)-(a^3)?

    I know that (2^15)-(2^3)=5*7*8*9*13
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  2. #2
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    Quote Originally Posted by MichaelG View Post
    Prove that (2^15)-(2^3) divides (a^15)-(a^3)?

    I know that (2^15)-(2^3)=5*7*8*9*13


    Hint: a^{15}-a^3=a^3\left(a^{12}-1\right)=a^3(a^6-1)(a^6+1)=a^3(a^3-1)(a^3+1)(a^6+1)= a^3(a-1)(a^2+a+1)(a^3+1)(a^6+1) ...

    Well, now just prove that the above is divisible by 5,7,8,9 and 13 no matter what a is.

    Tonio
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