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Math Help - Questions Involving Divisiblity

  1. #1
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    Questions Involving Divisiblity

    Hi,

    Could someone help me solve this, please.

    Show by Induction that for n >= 1:
    (3^(3n)) + (2^(n+2)) is divisible by 5

    Thanks.
    Last edited by mr fantastic; August 20th 2009 at 03:14 AM.
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  2. #2
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    Quote Originally Posted by justmaths View Post
    Hi,

    Could someone help me solve this, please.

    Show by Induction that for n >= 1:
    (3^(3n)) + (2^(n+2)) is divisible by 5

    Thanks.
    True for n = 1.

    Assume true for n = k.

    Show that under the assumption it's true for n = k + 1:

    3^{3(k+1)} + 2^{(k+1)+2} = 3^{3k} \cdot 3^3 + 2 \cdot 2^{k+2} = 27 \cdot 3^{3k} + 2 \cdot 2^{k+2}

    = 25 \cdot 3^{3k} + 2 \cdot 3^{3k} + 2 \cdot 2^{k+2} = 25 \cdot 3^{3k} + 2 \left( 3^{3k} + 2^{k+2} \right)

    and it should be clear how to finish things off.
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  3. #3
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    Oh I see. Then we let the last bracket equal a number divisible by 5.

    Thanks.
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