Let $n$ be a positive integer. Then $$ \sum_{d|n} \phi(d) = n$$

Where $\phi$ is the euler totient function.

The first step in a proof of this is to define a set $S_d$, for each divisor $d$ of $n$; $S_d={\{1 \leq a \leq n : gcd(a,n) = d}\}$.

It then states that it is "obvious" that $\sum_{d|n} |S_d| = n$, however this isn't remotely obvious to me?

Could anyone explain it to me please?

This has been flagged for a duplicate but the duplicate question doesn’t specifically tackle the step in the proof that is troubling me.