  Menon's identity is a classical identity involving gcd sums and the Euler
totient function $\phi$. In a recent paper, Zhao and Cao derived the Menon-type
identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)\chi(k) =
\phi(n)\tau(\frac{n}{d})$, where $\chi$ is a Dirichlet character mod $n$ with
conductor $d$. We derive an identity similar to this replacing gcd with a
generalization it. We also show that some of the arguments used in the
derivation of Zhao-Cao identity can be improved if one uses the method we
employ here.
