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 Post subject: A determinantPosted: Thu Mar 24, 2016 8:12 am
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Joined: Sat Nov 07, 2015 6:12 pm
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Location: Larisa
This post rang me also bells about a famous determinant.

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

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

Edit: Topic moved from "Linear Algebra" to "General Mathematics".

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