Integer Parities

If a,{\color{blue}{a}}, b,{\color{green}{b}}, and c{\color{red}{c}} are integers, is it possible that a+b,{\color{blue}{a}}+{\color{green}{b}}, a+c,{\color{blue}{a}}+{\color{red}{c}}, and b+c{\color{green}{b}}+{\color{red}{c}} are all odd numbers?


Note: It is possible for all three sums to be even!

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