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Math Help - Inclusion–exclusion principle

  1. #1
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    Inclusion–exclusion principle

    How many numbers that are less than 1000 000 are divisible by 7 but not divisible by 10,12,25?
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  2. #2
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    Re: Inclusion–exclusion principle

    Hello, Garas!

    How many numbers that are less than 1,000,000 are divisible by 7
    but not divisible by 10, 12, 25?

    Every 7th number is divisible by 7:. \left[\frac{999,999}{7}\right] \,=\,142,\!857

    But how many of these are also divisible by 10, 12, or 25?
    . . This will take a little work . . .


    \begin{Bmatrix}\text{Div by 7 and 10:} & \left[\dfrac{999,\!999}{7\cdot10}\right] &=& 14,\!285 \\ \\[-3mm] \text{Div by 7 and 12:} & \left[\dfrac{999,\!999}{7\cdot12}\right] &=& 11,\!904 \\ \\[-3mm] \text{Div by 7 and 25:} & \left[\dfrac{999,\!999}{7\cdot25}\right] &=& 5,\!714 \end{Bmatrix} \quad\text{Total: }31,\!903

    \begin{Bmatrix}\text{Div by 7, 10, 12:} & \left[\dfrac{999,999}{7\cdot60}\right] &-& 2,\!380 \\ \\[-3mm] \text{Div by 7, 12, 25:} & \left[\dfrac{999,999}{7\cdot300}\right] &=& 476 \\ \\[-3mm] \text{Div by 7, 10, 25:} & \left[\dfrac{999,999}{7\cdot50}\right] &=& 2,857 \end{Bmatrix} \quad\text{Total: }5,\!713

    \begin{Bmatrix}\text{Div by 7, 10, 12, 25:} & \left[\dfrac{999,\!999}{7\cdot300}\right] &=& 476 \end{Bmatrix} \quad\text{Total: } 476


    \text{There are: }\,31,\!903 - 5,\!713 + 476 \:=\:26,\!666\text{ multiples of 7}
    . . . . . . . . \text{which are also divisible by 10, 12, or 25.}


    \text{The answer is: }\:142,\!857 - 26,\!666 \;=\;116,\!191

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