Mathematical Contexts
Countless results in number theory invoke the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above choices have far-reaching consequences. For example, the fact that numbers have unique factorizations means that one can determine whether a number has an even or odd number of distinct prime factors. Since 1 is not prime, nor does it have prime factors, it is a product of 0 distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This implies that the Möbius function takes the value μ(1) = 1, which is necessary for it to be a multiplicative function and for the Möbius inversion formula to work, and affects the exact value of the Mertens function everywhere.
Read more about this topic: Parity Of Zero
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