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Math Help - Divisibility Problem

  1. #1
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    Divisibility Problem

    A list consists of 14 consecutive positive integers. Which of the following could be the number of integers in the list that are divisible by 13?

    Why would the answer be One or Two? How can I show that None is not the answer?
    Last edited by mjoshua; June 10th 2012 at 08:31 AM.
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  2. #2
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    Re: Divisibility Problem

    Quote Originally Posted by mjoshua View Post
    A list consists of 14 consecutive positive integers. Which of the following could be the number of integers in the list that are divisible by 13? Why would the answer be One or Two? How can I show that None is not answer?
    List the first fourteen: 1,~2,\cdots,~13,~14. What do you see?

    Now look at this list 13,~14,\cdots,~25,~26. What do you see?
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  3. #3
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    Re: Divisibility Problem

    The first list has a 13 and the second list has two numbers, 13 and 26, divisible by 13. So that's how we prove 1 or 2 numbers will be divisible by 13??? Also, how can we show that there will always be at least one number that will be?
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    Re: Divisibility Problem

    Quote Originally Posted by mjoshua View Post
    The first list has a 13 and the second list has two numbers, 13 and 26, divisible by 13. So that's how we prove 1 or 2 numbers will be divisible by 13??? Also, how can we show that there will always be at least one number that will be?
    Well it does not prove it. But help to see how things work.
    How 'far apart' are two multiples of thirteen?
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  5. #5
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    Re: Divisibility Problem

    Oh right, I mean that makes sense, but what does the proof look like for it?
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  6. #6
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    Re: Divisibility Problem

    You could try something like this:

    Suppose you have 14 consecutive positive integers i.e. n, n+1, n+2 ....... n+13.

    Case 1: n is divisible 13

    Then n+13 is divisible by 13 since \frac{n+13}{13}=\frac{n}{13} + 1 and we already assumed n is divisible by 13.

    So we proved we can have 2 factors, i.e n and n+13

    Case 2: n is not divisible 13

    Since n is not divisible by 13 we have,

    n = 13k + r where 0< r < 13 and k is an Integer
    n + r = 13k which implies that n+r is divisible by 13 where 0 < r < 13

    So 1 of your numbers from n+1, n+2.... ,n+12 is divisible by 13.

    Not sure whether this is good prove or not... but hopes it helps...
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  7. #7
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    Re: Divisibility Problem

    You can use a Pigeonhole-type argument to claim that "none" cannot be the answer.

    "1 or 2" works. Consider the following sequences of 14 integers:


    1,2,...,14
    13,14,...,26
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