Let us take an integer x which is both even

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.

*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.

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