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Math Help - value of N

  1. #1
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    value of N

    N is the number of buttons in a sewing box.
    a). N is more than 40 but less than 80.
    b).When N is divided by 5, the remainder is 2.
    c).When N is divided by 7, the remainder is 4.
    Find the value of N.
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  2. #2
    Super Member Bacterius's Avatar
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    So you have 40 < n < 80.
    \Rightarrow The set of numbers is {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79}.

    When n is divided by 5, the remainder is two, so n \equiv 2 \pmod{5}. Thus n = 2 + 5k with k \in Z.
    \Rightarrow The set of numbers is {2, 7, 12, ... 47, 52, 57, 62, 67, 72, 77, ...}

    When n is divided by 7, the remainder is four, so n \equiv 4 \pmod{7}. Thus n = 4 + 7k with k \in Z.
    \Rightarrow The set of numbers is {4, 11, 18, ... 46, 53, 60, 67, 74, ...}

    Now intersect those three sets : which numbers are in all three of them ?

    Our three sets are :

    {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79}

    {2, 7, 12, ... 47, 52, 57, 62, 67, 72, 77, ...}

    {4, 11, 18, ... 46, 53, 60, 67, 74, ...}

    In red, the numbers that are in the first set.
    And in blue, the number that is in all three sets.
    \rightarrow Thus, n = 67.
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  3. #3
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    thank u for ur help!
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