Smith’s determinant

Let \gcd(i,j) denote the greatest common divisor of i, j and \varphi the Euler’s totient function. Prove that:

\displaystyle \begin{vmatrix} \gcd(1,1) &\gcd(1, 2) &\cdots & \gcd(1,n)\\ \gcd(2,1)&\gcd(2,2) &\cdots & \gcd(2,n)\\ \vdots& \vdots & \ddots &\vdots \\ \gcd(n,1)&\gcd(n,2) &\cdots &\gcd(n,n) \end{vmatrix}= \prod_{j=1}^{n}\varphi(j)

Solution

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