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Math Help - Math egg puzzle Help

  1. #1
    Junior Member Dragon's Avatar
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    Math egg puzzle Help

    a peddler is taking edge to the market to sell. The eggs are in a cart that can hold 500 eggs. If the eggs are removed fom the cart 2 3 4 5 or 6 at a time 1 egg s alwas left over If the eggs are removed 7 at a time no eggs are left over how many eggs are their in the cart?
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  2. #2
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    Hello, Dragon!

    A peddler is taking eggs to the market to sell.
    The eggs are in a cart that can hold 500 eggs.
    If the eggs are removed fom the cart 2, 3, 4, 5 or 6 at a time,
    . . one egg is always left over.
    If the eggs are removed 7 at a time, no eggs are left over.
    How many eggs are their in the cart?
    Let N be the number of eggs.

    The LCM of 2, 3, 4, 5 and 6 is 60.
    . . Hence, 60 (and any multiple of 60) is divisible by 2, 3, 4, 5 and 6.
    To have a remainder of 1, N must be of the form: . 60k + 1
    . . for some positive integer k.
    That is: N \:=\:60k + 1 [1]

    Since N is divisible by 7\!:\;\;60k + 1 \:=\:7a


    If you're familiar with Modulo Arithmetic: . 60k \,\equiv \,-1\!\! \pmod{7}

    Reduce: . 4k \,\equiv \,6\!\! \pmod{7}

    Multiply by 2: . 8k\,\equiv\,12\!\!\pmod{7}\quad\Rightarrow\quad k\,\equiv\,5\!\!\pmod{7}

    Hence, k is of the form: 7b + 5 for some integer b.
    That is: k \:=\:7b + 5 [2]


    Substitute [2] into [1]: . N \:=\:60(7b + 5) + 1\:=\:420b + 301

    Since 0 < N \leq 500, then b = 0

    Therefore: . \boxed{N \,= \,301}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    If you're not familiar with Modulo Arithmetic,
    . . it can be solved with "ordinary" algebra.
    It just takes longer.

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