Prove there is no integer that is both even and odd

Discussion in 'Calculator Requests' started by math_celebrity, Jan 27, 2024.

  1. math_celebrity

    math_celebrity Administrator Staff Member

    Let us take an integer x which is both even and odd.
    • As an even integer, we write x in the form 2m for some integer m
    • As an odd integer, we write x in the form 2n + 1 for some integer n

    Since both the even and odd integers are the same number, we set them equal to each other
    2m = 2n + 1

    Subtract 2n from each side:
    2m - 2n = 1

    Factor out a 2 on the left side:
    2(m - n) = 1

    By definition of divisibility, this means that 2 divides 1.

    But we know that the only two numbers which divide 1 are 1 and -1.

    Therefore, our original assumption that x was both even and odd must be false.

    Last edited: Jan 27, 2024

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