I've recently come upon the following (seemingly) simple observation:
Claim: A positive integer $k = d d^{\prime} > 1$ has the opposite parity of $\text{gcd}(d+1,d^{\prime}+1)$ for any pair of distinct divisors $d$ and $d^{\prime}$ of $k$.
There are substantially simple proofs (see comments), and one proof follows from the formula: \begin{eqnarray} \text{gcd}(a,b) = a + b - ab + 2 \sum_{k=1}^{a-1} \lfloor \tfrac{kb}{a} \rfloor. \end{eqnarray}
Corollary: Given a multiplicative arithmetic function $f \colon \mathbb{N} \to \mathbb{N}$ and a pair of coprime integers $a$ and $b$, \begin{eqnarray} f(ab) + 1 \equiv \text{gcd}(f(a)+1,f(b)+1) \mod 2. \end{eqnarray}
Question: Do either of these simple results have any neat use(s) or interesting application(s)?
(NB: The purpose of this post is not so much about finding simple (or simpler) proofs, but rather more about determining possible applications.)
Thanks!