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Math Help - Sequences and series

  1. #1
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    Sequences and series

    Dear math forum members,

    I have a problem


    How many integers divisble by 7 can you find in the interval [4500, 7000] ?


    Could you please give me just a hint on how to proceed?



    Thank you in advance!
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  2. #2
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    Hello,

    Firstly, i'll note that 7000 is a multiple of 7.

    So we'll start from there.

    The previous multiple of 7 will be obtained by substracting 7 to 7000, and so on.

    So if you divide the interval into intervals of 7 numbers, ]a;b], these interval will contain one and only one multiple of 7.

    Hence you "just" have to see how many intervals of this sort there are in [4500;7000], and this number is given by (7000-4500)/7.
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  3. #3
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    Except, that the number comes out as a decimal...
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  4. #4
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    I know :-)

    But it's as if you could restrict your study to the interval [first multiple of 7 coming after 4500 = N; 7000]

    The number of multiples of 7 in [N;7000] will be (7000-N)/7 + 1 (because you count the extremity).
    And this number is the same as the number of multiples of 7 in [4500;7000] because there is no multiple of 7 between 4500 and N.

    So this shows that if you truncate (7000-4500)/7 by the inferior value (because (7000-4500)/7 = (7000-N)/7 + APositiveNumber) and add 1, you'll have the number you want.
    This is the same as taking the superior integer of (7000-4500)/7

    I don't know if it's clear enough

    You can take a smaller example, such as [10;35]
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  5. #5
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    Hello, Coach!

    How many integers divisble by 7 can you find in the interval [4500, 7000] ?

    Since every seventh number is divisible by 7,
    . . there are: . \frac{7000}{7} \:=\:1000 of them on the interval [0, 7000]

    We must eliminate the multiple that are less than 4500.
    How many are there?
    . . There are: . \frac{4500}{7} \:=\:642.857...\:\Rightarrow\:642 of them.


    Therefore, there are: . 1000 - 642 \:=\:358 multiples of 7 on [4500, 7000]

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  6. #6
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    Coach, do you know the floor function (aka: the greatest integer function)?
    The floor of x, \left\lfloor x \right\rfloor , equals the largest integer which does exceed x.
    Some examples: \left\lfloor \pi  \right\rfloor  = 3\,,\,\left\lfloor { - e} \right\rfloor  =  - 3\,\& \,\left\lfloor 3 \right\rfloor  = 3.
    Most calculators have such a function built in.

    For positive integers d\,\& \,N,\,d < N the number of multiples of d in [1,N] is \left\lfloor {\frac{N}{d}} \right\rfloor .

    Thus to use a calculator to solve your problem we would calculate:
    \left\lfloor {\frac{{7000}}{7}} \right\rfloor  - \left\lfloor {\frac{{4499}}<br />
{7}} \right\rfloor .
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