# Re: Proof that the sum of two consecutive integers is always an odd number.

*From*: Stan <google@xxxxxxxxxxxxxxx>*Date*: Sat, 10 Nov 2007 00:38:30 -0800

On 10 Nov, 01:30, Kenneth Doyle <nob...@xxxxxxxxxxx> wrote:

I saw this last week as a question here. I've never been good at proofs,

but for some reason a "sketch" of this proof just popped into my head.

We want to prove that the sum of two consecutive integers is always an

odd number, that is:

x + y = n, where y = x + 1 and n is an odd number.

This can be re-written as:

x + x + 1 = n, because y = x + 1.

or:

2x + 1 = n.

Any integer multiplied by two is an even number and an even number plus

one is an odd number, therefore the sum of two consecutive integers is

always an odd number.

In this example, would I also have to prove that an integer multiplied by

two is always even? Something like:

2x = m, where m is even, because m / 2 always gives us

the original integer x and a number is even if it can be divided by two

leaving no remainder.

Would the proof that m is always even, have to be included in the proof

or can it be just a reference to a "sub-proof"; if you see what I mean.

Sorry about the clumsy language.

Try formulating the definition of an even number. Or ask yourself,

what is an even number if not a multiple of 2?

.

**References**:**Proof that the sum of two consecutive integers is always an odd number.***From:*Kenneth Doyle

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