Even and odd split the integers by divisibility by two
Parity asks whether an integer is even—divisible by 2—or odd. −4, 0, and 82 are even; −7, 5, and 23 are odd. Zero is even; consecutive integers always flip parity. Fractions like 1/2 fall outside the basic definition.
In base ten the last digit settles the question: numbers ending in 1, 3, 5, 7 or 9 are odd, those ending in 0, 2, 4, 6 or 8 even. Binary works the same way with a final 0 or 1, and so does any even base. In an odd base the test switches to the digit sum, so in base seven the numeral 11, worth 8, counts as even. When one integer divides another exactly, the quotient is even only if the dividend carries more factors of two than the divisor.
Abstractly, the evens form an ideal in the ring of integers while the odds do not, since zero belongs only to the evens. Arithmetic modulo 2 turns the pair even and odd into the field with two elements, where subtraction behaves exactly like addition, and parity checks offer a quick way to catch a wrong equation. The ancient Greeks regarded 1, the monad, as neither fully odd nor fully even, and Friedrich Fröbel's 1826 The Education of Man still told teachers to drill that idea into pupils.
Every prime except 2 is odd, while every known perfect number is even and nobody knows whether an odd one exists. Goldbach's conjecture, that each even integer above 2 is a sum of two primes, has been confirmed by computer up to at least 4 × 10^18 without a general proof. The Feit–Thompson theorem makes every finite group of odd order solvable. Beyond pure mathematics, a parity bit is the simplest error-detecting code, clarinets sound only odd harmonics, and in the United States even-numbered highways mostly run east to west.
Source: Parity (mathematics)
