Expressions that are always even

Parity rules let you answer questions without doing the sum. Is 672 − 348 even? Both numbers are even, and even − even = even, so yes, without subtracting anything.

The same reasoning works on algebraic expressions, and there it settles not one case but all of them at once.

Take 4m + 2q. Is it even for every pair of integers m and q? Two ways to see it:

• First: 4m is even whatever m is, and 2q is even whatever q is. Even + even = even. Done.
• Second, often neater: 4m + 2q = 2(2m + q). Written that way, 2 is plainly a factor of the whole thing, so whatever is in the bracket, the result is even. Check with m = 4 and q = −9: 4 × 4 + 2 × (−9) = 16 − 18 = −2, which is even.

Now one that isn't always even: x² + 2. If x is even, x² is even and x² + 2 is even: x = 6 gives 38. If x is odd, x² is odd and x² + 2 is odd: x = 3 gives 11. One example and one non-example settle it: x² + 2 is not always even.

Remember the even-number test while you're here: add or subtract 1 from any even number and you get an odd one. 0 + 1 = 1 and 0 − 1 = −1 are both odd, so 0 is even. (So is a − a, for any integer a.)

That's the useful imbalance. To show an expression is always even you need an argument covering every case. To show it is not, a single non-example is enough.