# 2022-21 A determinant of greatest common divisors

Let $$\varphi(x)$$ be the Euler’s totient function. Let $$S = \{a_1,\dots, a_n\}$$ be a set of positive integers such that for any $$a_i$$, all of its positive divisors are also in $$S$$. Let $$A$$ be the matrix with entries $$A_{i,j} = gcd(a_i,a_j)$$ being the greatest common divisors of $$a_i$$ and $$a_j$$. Prove that $$\det(A) = \prod_{i=1}^{n} \varphi(a_i)$$.

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