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Prove that among any 3 integers there are always two whose difference is divisible by 2.

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HINT: Is the difference of two even numbers divisible by $2$? What about the difference of two odd numbers?

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Assume your integers are $x_1<x_2<x_3$. If both $x_2-x_1$ and $x_3-x_2$ are odd, then $x_3-x_1=(x_3-x_2)+(x_2-x_1)$ is a sum of two odd numbers and therefore even.

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