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Math Help - I feel like an idiot

  1. #1
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    I feel like an idiot

    Am I missing something obvious, or is this really just exhaustion method?

    "Find the smallest positive integer K which is divisible by seven and which, divided by each of the integers between two and six inclusive, leaves a remainder of one. If possible, find a larger value of K."
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Hiphopopotamus View Post
    Am I missing something obvious, or is this really just exhaustion method?

    "Find the smallest positive integer K which is divisible by seven and which, divided by each of the integers between two and six inclusive, leaves a remainder of one. If possible, find a larger value of K."
    Hmmm do you know the Chinese remainder theorem ?

    Note that if the remainder in the division by 4 is 1, then it is 1 in the division by 2 too. Because K=4k+1=2*(2k)+1. So you don't need to bother yourself working on 2.
    Same goes for K=6k'+1=3*(2k)+1



    Edit : never say you're an idiot.
    Last edited by Moo; September 8th 2008 at 10:18 AM.
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  3. #3
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    You cannot be an idiot with the screenname 'hiphopopotamus'.
    Go Flight of the Conchords!

    Just thought I'd say that. Sorry I can't help with your question.
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  4. #4
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    Hello, Hiphopopotamus!

    Find the smallest positive integer K which is divisible by 7,
    and which, divided by each of the integers between 2 and 6 inclusive, leaves a remainder of one.
    If possible, find a larger value of K.
    See if this makes sense to you . . .


    The Lowest Common Multiple of {2, 3, 4, 5, 6} is 60.

    Then K is of the form: . K \;=\;60m + 1

    Since K is divisible by 7: . K \:=\:7n


    And we have: . 7n \:=\:60m + 1 \quad\Rightarrow\quad n \:=\:\frac{60m+1}{7} \quad\Rightarrow\quad n \:=\:8m + \frac{4m+1}{7}

    Since n is an integer, 4m+1 must be divisible by 7.
    . . The first time this happens is when m = 5.
    . . The next time this happens is when m = 12.


    Therefore, the smallest value is: . K \;=\;60(5) + 1 \;=\;\boxed{301}

    And the next value is: . K \;=\;60(12) + 1 \;=\;\boxed{721}

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