The sum of any four consecutive integers is never divisible by 4

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I want to prove this statement:

S : The sum of any four consecutive integers is never divisible by 4.

Proof: The sum of four consecutive integers $n,n+1,n+2,n+3$ is $4n + 6$.

If I divide $4n + 6$ by $4$, the remainder is always $2$.

Hence S is true.

Is the proof correct?

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1 Answer

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Yup, that's right. Your integers are $n, n+1, n+2, n+3$. You add them up to get $4n+6$. If you don't want to talk about remainders or congreuences, then you can also say that $\frac{4n+6}4 = n + \frac {6}{4} = n + 1 + \frac{1}{2}$. And $n+1+\frac 1 2$ is clearly not an integer, so $4n+6$ is not divisible by $4$.

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