Definition of factor - Is n a factor of n?

Question

Is there a universally agreed upon definition of what a factor of a number is?

Is $n$ a factor of $n$?

Is $1$ a factor of $n$?

EDIT x 2: Integers Natural Numbers

Answer 1

It is meaningless to talk about factors without agreeing in advance what "numbers" we are using. For example, $3$ is not a factor of $5$ if we use integers, but it is a factor of $5$ if we use fractions, because $3\cdot \frac{5}{3}=5$.

With this understanding, we say that

$a$ is a factor of $b$ if there is another "number" $c$ (of the agreed-upon type), such that $ac=b$.

Because $1\cdot n=n$, both $1$ and $n$ are factors of $n$ (so long as our agreed-upon numbers contain $1$).

Answer 2

Yes, $1$ and $n$ are, trivially, factors or divisors of $n$. In particular, if $n$ is a prime number, then the prime factorization of $n$ is simply $n$ itself.

That said, when speaking of the factors of a number, it is often useful to exclude these trivial factors. The usual term for that is "proper factor" or "proper divisor". Specifically, a proper divisor of a number $n$ is normally defined to be any divisor of $n$ which is strictly greater than $1$ and less than $n$. In particular, this means that prime numbers do not have proper divisors, whereas all composite numbers do.