The sum of one odd number and one even number is another odd number.
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Any pair of one even and one odd number will have an even product and an odd sum.
The sum of an odd and even number is always odd because there's always one left over. Think of any odd number as being any even number plus one. If you add two even numbers, the answer is always even. If you add an even and an odd (plus one), the answer is always odd.
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The sum of two odd numbers is always even.
If you're adding an odd number of numbers, if two are even and one is odd, you will get an odd number as a result. Three odds are odd. If two are odd and one is even, you have an even result. Three evens are even.