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Math Help - Divisibility

  1. #1
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    Divisibility

    Hi all.

    I'm trying to figure out the following problem:

    Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4.

    Help will be appreciated. Looking for a simple/elementary proof.

    Thanks.
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by pollardrho06 View Post
    Hi all.

    I'm trying to figure out the following problem:

    Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4.

    Help will be appreciated. Looking for a simple/elementary proof.

    Thanks.
    Hint:
    |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3 \text{ but not by } 4\}|
     = |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3\}|-|\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 12\}|

    This is a consequence of A\backslash B=A\backslash(A\cap B), and |X\backslash Y|=|X|-|Y|, if Y\subseteq X.
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  3. #3
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    Quote Originally Posted by Failure View Post
    Hint:
    |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3 \text{ but not by } 4\}|
     = |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3\}|-|\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 12\}|

    This is a consequence of A\backslash B=A\backslash(A\cap B), and |X\backslash Y|=|X|-|Y|, if Y\subseteq X.
    I don't see it...
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  4. #4
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    Hint

    Look for cycles.
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  5. #5
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    Hello, pollardrho06!

    Find the number of positive integers not exceeding 1000
    that are divisible by 3 but not by 4.

    Every third number is divisible by 3.
    . . There are: . \left[\frac{1000}{3}\right] \:=\:333 numbers divisible by 3.

    But every twelfth number is divisible by 3 and by 4.
    . . There are: . \left[\frac{1000}{12}\right] \:=\:83 multiples of 3 which are divisible by 4.


    Therefore, there are: . 333 - 83 \:=\:250 such numbers.

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  6. #6
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    Quote Originally Posted by Soroban View Post
    Hello, pollardrho06!


    Every third number is divisible by 3.
    . . There are: . \left[\frac{1000}{3}\right] \:=\:333 numbers divisible by 3.

    But every twelfth number is divisible by 3 and by 4.
    . . There are: . \left[\frac{1000}{12}\right] \:=\:83 multiples of 3 which are divisible by 4.


    Therefore, there are: . 333 - 83 \:=\:250 such numbers.

    Wow!! The greatest integer function!! Gr8!! Thanks!!
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