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There is something missing from your question: what about the two consecutive integers? Is the proof required that one of them is divisible by 2? Or that their product is (which amounts to the same thing)?

Showing that one of two consecutive number is divisible by 2:

Suppose your two numbers are n and (n+1).

If n is divisible by 2, ie n = 2k, the result is shown.

Otherwise assume n is not divisible by 2. In this case n = 2m+1. Then:

(n+1) = ((2m+1)+1)

= 2m + 2

= 2(m+1)

which is a multiple of 2 and so divisible by 2. QED.

Showing that exactly one of two consecutive integers is divisible by two is shown above with the addition to the first part:

"as (n+1) = 2k+1 is not divisible by two and so only n is divisible by 2."

To show the product is divisible by 2, show either n is divisible by 2 or (n+1) is as above, then the result follows as one of n and (n+1) is divisible by 2 and so their product is.

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Q: Prove that the two consecutive positive integer is divisible by 2?
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