# Why is $1$ a proper divisor?

Intuitively, the proper divisors of an integer $n$ don't include $n$ because trivially, any number divides itself; but $1$ divides any integer as well.

What is the rationale for including $1$?

• Proper divisor of a number $n$ means a divisor $d$ smaller than the original number $n$. This definition is diferent from being a trivial divisor. Every number $n$ has two trivial divisors: $n$ itself and $1$. – Xam Oct 28 '16 at 1:57

Similarly, you can say that we count $1$ as relatively prime to all integers so that the totient function (number of integers less than $n$ which are relatively prime to $n$) will be multiplicative. That is, if $m$ and $n$ are relatively prime, then $$\varphi(mn) = \varphi(m) \varphi(n)$$