Integer Parities

If a,{\color{#3D99F6}{a}}, b,{\color{#20A900}{b}}, and c{\color{#D61F06}{c}} are integers, is it possible that a+b,{\color{#3D99F6}{a}}+{\color{#20A900}{b}}, a+c,{\color{#3D99F6}{a}}+{\color{#D61F06}{c}}, and b+c{\color{#20A900}{b}}+{\color{#D61F06}{c}} are all odd numbers?


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

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