Integer Parities

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

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

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