6. (8) Which natural numbers can be represented in the form $[x, y]+[y, z]+[z, x]$ with natural $x, y, z$? $[a, b]$ is the least common multiple of numbers $a$ and $b$.

保留了源文本的换行和格式。
6. Answer: All, except powers of 2.

First, consider a number that is not a power of two, $n=2^{p}(2 q+1), q>0$. We have $n=2^{p}+2^{p} q+2^{p} q=$ $\left[1,2^{p}\right]+\left[1,2^{p} q\right]+\left[2^{p}, 2^{p} q\right]$. Now let's prove that a power of 2 cannot be represented in such a form. If $x=2^{a} u, y=2^{b} v, z=2^{c} w$, $a \leqslant b \leqslant c, u, v, w$ are odd, then $[x, y]+[y, z]+[z, x]=2^{b}\left([u, v]+2^{c-b}([v, w]+[u, w])\right)$. The expression in parentheses is odd and greater than 1, so the entire product cannot be a power of 2.