A determinant (1)
- Tolaso J Kos
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A determinant (1)
Let $(a, b)$ denote the greatest common divisor of $a, b$. Prove that:
$$\begin{vmatrix}
\left ( 1,1 \right ) &\left ( 1,2 \right ) &\cdots &\left ( 1,n \right ) \\
\left ( 2,1 \right )&\left ( 2,2 \right ) & \cdots &\left ( 2,n \right ) \\
\vdots & \vdots & \ddots & \vdots \\
\left ( n,1 \right )& \left ( n, 2 \right ) &\cdots & \left ( n,n \right )
\end{vmatrix} = \prod_{k=1}^{n}\phi(k)$$
where $\phi$ is Euler's $\phi$ function.
$$\begin{vmatrix}
\left ( 1,1 \right ) &\left ( 1,2 \right ) &\cdots &\left ( 1,n \right ) \\
\left ( 2,1 \right )&\left ( 2,2 \right ) & \cdots &\left ( 2,n \right ) \\
\vdots & \vdots & \ddots & \vdots \\
\left ( n,1 \right )& \left ( n, 2 \right ) &\cdots & \left ( n,n \right )
\end{vmatrix} = \prod_{k=1}^{n}\phi(k)$$
where $\phi$ is Euler's $\phi$ function.
Imagination is much more important than knowledge.
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