Equal parities add to even; different parities add to odd. A product is odd only when every factor is odd.
Addition and subtraction
For integers, there are just three addition cases:
| Calculation | Result | Example |
|---|---|---|
| Even + even | Even | 14 + 8 = 22 |
| Even + odd | Odd | 14 + 9 = 23 |
| Odd + odd | Even | 15 + 9 = 24 |
Subtraction follows the same parity pattern: equal parities give an even difference, and different parities give an odd difference. The difference can be negative without changing the rule. For example, 5 − 12 = −7 is odd.
A useful mental shortcut is that adding or subtracting an even integer preserves parity. Adding or subtracting an odd integer flips it. This follows directly from treating parity as the two remainder classes discussed in the SUNY Schenectady congruence lesson.
Why two odd integers add to even
Write the odd integers as 2a + 1 and 2b + 1, with integer values of a and b. Their sum is:
(2a + 1) + (2b + 1) = 2(a + b + 1).
The quantity inside the parentheses is an integer, so the sum is twice an integer. That proves the rule for every pair of odd integers, including negative ones.
For several terms, pair up the odd terms. An even count of odd terms gives an even sum; an odd count gives an odd sum. Any even terms can be ignored when determining only parity.
Multiplication and powers
A product with an even factor is even because that factor already contains a factor of two. A product of two odd integers is odd:
(2a + 1)(2b + 1) = 2(2ab + a + b) + 1.
Repeated multiplication extends the rule: every positive integer power of an odd integer is odd. Every positive integer power of an even integer is even. For a nonzero base, the zeroth power is 1, so the positive-exponent condition matters.
Division does not fit a simple table
An even integer divided by an even integer can produce either parity: 18 ÷ 2 = 9, while 20 ÷ 2 = 10. It can also produce a non-integer, as in 6 ÷ 4 = 1.5. Check the actual quotient before classifying it.
Try predicting 31 + 44 + 17 without calculating the total. Two odd terms combine to even, and the remaining term is even, so the sum is even. Its total, 92, confirms the prediction. Use the calculator to check a resulting integer, or review the odd-integer definition behind these proofs.