= Solution
The <habitable zone> is the range of orbital distances where an appropriate planetary atmosphere could sustain liquid water at the surface; it is not a guarantee of inhabited or even habitable conditions for every planet. At a fixed habitable incident <flux>, $a_{\rm HZ}\propto L_\star^{1/2}$. With the supplied $L_\star\propto M_\star^3$, <Kepler's third law> gives
$$
\boxed{a_{\rm HZ}\propto M_\star^{3/2},\qquad P_{\rm HZ}\propto a_{\rm HZ}^{3/2}M_\star^{-1/2}\propto M_\star^{7/4}.}
$$
Taking Earth as a rough central reference, a $0.1M_\odot$ star has $a_{\rm HZ}\sim0.0316\,\mathrm{AU}$ and $P_{\rm HZ}\sim6.5$ days. Its six-day, Earth-sized companion receives about $1.11$ times terrestrial <irradiance> and could therefore lie in the <habitable zone> in this simplified model. Atmosphere, stellar activity, tidal locking and volatile retention remain unknown. The six-day companion around the Sun-like star receives about $239$ times terrestrial <irradiance> and is approximately Jupiter-sized; it is not a plausible surface-water Earth analogue. The stellar spectral dependence of actual habitable-zone edges is omitted by the simple scaling.
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