Is 0 an odd number or an even number?

Zero is definitively an even number. This classification is not a matter of convention but a direct, logical consequence of the formal mathematical definition of evenness. An integer is defined as even if it is divisible by 2 with no remainder, or equivalently, if it can be expressed in the form 2k, where k is an integer. Zero satisfies this definition perfectly: 0 divided by 2 equals 0, which is an integer with no remainder, and 0 can be written as 2 multiplied by 0 (since 2 * 0 = 0). Any integer that is not even is, by definition, odd, meaning it can be expressed as 2k+1 for some integer k. There is no integer k for which 2k+1 equals 0, which conclusively places zero outside the set of odd numbers. This foundational arithmetic principle is consistent across all standard mathematical systems, from basic number theory to advanced algebra.

The evenness of zero is further reinforced by its behavior within the broader structures of mathematics. In modular arithmetic, an integer is even if it is congruent to 0 modulo 2. Zero modulo 2 is unequivocally 0, aligning it with the even numbers. The parity rules for addition and multiplication also depend on zero being even. For instance, the sum of two even numbers is always even; since 2 and -2 are even, their sum is 0, which must therefore be even for the rule to hold universally. Similarly, an even number multiplied by any integer yields an even product; if zero were not even, this fundamental rule would require an exception, creating unnecessary complexity and inconsistency. Its role as the additive identity—the number that, when added to any other, leaves it unchanged—fits seamlessly within the even number set, as adding an even number does not change the parity of the other operand.

Some historical or pedagogical confusion about zero's parity occasionally arises from informal, pattern-based introductions to even and odd numbers, such as grouping objects into pairs. Visualizing "zero objects" can seem abstract, leading to the mistaken notion that zero is somehow neither even nor odd. However, this is a conceptual hurdle, not a mathematical ambiguity. In set theory and the study of number systems, zero's status is unambiguous. Its classification as even is essential for theorems like the parity of squares: every integer squared is even if and only if the integer itself is even. Since 0 squared is 0, and 0 is even, the theorem holds without special case. This consistency is critical for automated theorem provers and computer algebra systems, where zero must have a defined parity for logical operations to function correctly.

Therefore, the answer is absolute: zero is an even integer. This conclusion is derived from core definitions, preserved by algebraic rules, and required for mathematical coherence. There is no reputable mathematical framework that treats zero as odd or as a number without parity. Any alternative interpretation would introduce fractures in arithmetic properties and combinatorial reasoning, making zero a disruptive exception rather than the integrative element it is within the number system. Its evenness is a settled, fundamental fact in mathematics.