Let \(n\ge 1\) and let \(\mathbb {Z}_n^\star \) be the group of units in the ring of residual classes modulo n, let \(m\ge 0\), \(k\ge 0\), \(m+k\ge 1\), \(u_1,\ldots ,u_m\in \mathbb {Z}_n^\star \) and \(S_1, \ldots , S_m\) nonempty subsets of \(\mathbb {Z}_n^\star \). In this note we shall explicitly compute the following sum \(\sum _{\begin{array}{c} a_1\in S_1, \ldots , a_m\in S_m \\ b_1,\ldots , b_k\in \mathbb {Z}_n \end{array}}gcd( a_1-u_1,\ldots , a_m-u_m, b_1,\ldots ,b_k, n).\) Moreover, for a nonempty subset \(S\subset \mathbb {Z}_n^\star \) and any polynomial f with integer coefficients we compute the sum \(\sum _{t\in S}gcd(f(t), n).\) This generalizes a well-known Menon-type identity with polynomials.

令\（n \ ge 1 \）和\（\ mathbb {Z} _n ^ \ star \）是模数为n的剩余类环中的单位组，令\（m \ ge 0 \），\（ k \ ge 0 \），\（m + k \ ge 1 \），\（u_1，\ ldots，u_m \ in \ mathbb {Z} _n ^ \ star \）和\（S_1，\ ldots，S_m \）\（\ mathbb {Z} _n ^ \ star \）的非空子集。在本说明中，我们将显式计算以下总和\（\ sum _ {\\ begin {array} {c} a_1 \ in S_1，\ ldots，a_m \ in S_m \\ b_1，\ ldots，b_k \ in \ mathbb {Z } _n \ end {array}} gcd（a_1-u_1，\ ldots，a_m-u_m，b_1，\ ldots，b_k，n）。\）此外，对于非空子集\（S \ subset \ mathbb {Z} _n ^ \ star \）以及任何具有整数系数的多项式f，我们计算总和\（\ sum _ {t \ in S} gcd（f（t），n）。\）这将公知的Menon型恒等式推广为多项式。