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The sum of an odd and an even number is odd.

Any odd number can be expressed as 2n + 1 (for some integer "n").

Any even number can be expressed as 2m (for another integer, "m").

The sum of the two is 2(m+n) + 1. Since the expression in parentheses is an integer, multiplying it by 2 gives you an even number. Adding 1 makes the entire expression odd.

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