You are watching: Why is 1 neither prime nor composite

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Anyway, let"s reduced to the chase. Exactly how come $1$ isn"t a prime number or even a composite number? I want to recognize from her great-heard-of answers! At least I understand that a element number has actually only two factors, $1$ and itself. So, why isn"t this the very same for $1$? It has actually a element of $1$, i beg your pardon is additionally itself, $1$. I want to hear around this, too.

Because if it to be prime, climate the prime factorization of numbers wouldn"t it is in unique. Therefore it"s identified as not-prime.

Because $1$ is a unit in $Bbb Z$. In every ring, in details in every UFD you have actually units and nonunits. It is the nonunits that space factored right into irreducible (=prime) components times units, yet one doesn"t element units. Unique factorization is distinct up to unit multipliers, because one can add an arbitrary unit times its inverse, and also still acquire a great factorization.

The main valuable idea behind primes is to get rid of all your multiples as composites, watch sieve the Eratosthenes. What would take place if us were to apply this reasoning to the number $1$ ?

For a number to be a element number it must have actually two factors one and it"s self, but with 1 just 1 time one is 1, so it only has actually one factor, therefore making it not a element number.

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