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    • Well, 2x is always divisible by 2. (One thing to be careful about, is what kinds of thing x can be. Here I am allowing x to be any non-negative integer: 0, 1, 2, 3 ...)

    And 2x+1 is never divisible by 2 (where x is again a non-negative integer).

    This is a perfectly abstract way of writing even and odd numbers.

    Now the sum of an even number and an odd number can be written in perfect generality as:
    2x + (2y+1)
    where x and y are non-negative integers. (Why did I have to use both an x and a y?)

    Then,

    2x + (2y+1) = 2(x + y) + 1
    which must be odd!