Prove that out of two consecutive integers, one is divisible by 2

Hi. I'm supposed to prove that given any two consecutive integers a and a+1, one of them is divisible by 2. And I'm supposed to use the Division Algorithm. I'm not entirely sure where to begin.

Any help is appreciated!

Re: Prove that out of two consecutive integers, one is divisible by 2

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Originally Posted by

**oregongirl** Hi. I'm supposed to prove that given any two consecutive integers a and a+1, one of them is divisible by 2. And I'm supposed to use the Division Algorithm. I'm not entirely sure where to begin.

Any help is appreciated!

suppose a is divisible by 2. You're done.

suppose a is not divisible by 2. Then a = 2k + 1 for some k (i.e. a is an odd number)

then a + 1 = (2k+1) + 1 = 2k + 2 = 2(k+1) and thus (a+1) is divisible by 2