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show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is any positive integer.

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    The positive integers +1, +3 and +5 are not of that form, unless q can also be zero.

    The difference between 6q + 5 and 6q' +1, if q' = q+1 , is
    [6(q+1) + 1]-[6q +5] = 2
    That means that by increasing q by 1 when you get to 6q+5, the next number will be 2 greater, so that all odd numbers can be created. All one has to do is pick the next integer q when you get to 6q +5, and start over at 6q' +1

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