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Math Help - Proof

  1. #1
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    Proof

    Prove that if is greater than and divisible by , but not divisible by , then:

    is the remainder when is divided by where is a natural number.
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  2. #2
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    Quote Originally Posted by usagi_killer View Post
    Prove that if is greater than and divisible by , but not divisible by , then:

    is the remainder when is divided by where is a natural number.
    \left(\frac{n}{2} + 1\right)^k - \left(\frac{n}{2} + 1\right) =  \left(\frac{n}{2} + 1\right) \left\{\left(\frac{n}{2} + 1\right)^{k-1} - 1\right\} = \left(\frac{n}{2} + 1\right) \times \frac{n}{2} \times (\text{some number})

    It suffices to show \left(\frac{n}{2} + 1\right) \times \frac{n}{2} = \frac{n(n+2)}{4} is divisible by n. But observe that n is an even number not divisible by 4, thus 4 | n+2 and hence \frac{n(n+2)}{4} is divisible by n and so is the original expression.
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