Smith (1875) evaluated the determinant of the $N \times N$ matrix with entries $\gcd(i, j)$, and more generally of $f(\gcd(i,j))$ when $f = g * 1$ is a Dirichlet convolution: it equals $\prod_{k \le N} g(k)$. The matrix of least common multiples is not of that form, but $\mathrm{lcm}(i,j)\gcd(i,j) = ij$ writes it as $D\,S\,D$ with $D = \mathrm{diag}(1, \dots, N)$ and $S_{ij} = 1/\gcd(i,j)$, and $1/m$ is a Dirichlet convolution $g * 1$ with $g = \mu * (1/\cdot)$ by Möbius inversion. Hence

so that the determinant is $N! \prod_{k \le N} \prod_{p \mid k} (1 - p)$. The three statements are the determinant in terms of $g$, the closed form of $g$, and the closed form of the determinant.