We present synthetic far- and near-ultraviolet (FUV and NUV) maps of M31, both with and without dust reddening. These maps were constructed from spatially-resolved star formation histories (SFHs) derived from optical Hubble Space Telescope imaging of resolved stars, taken as part of the Panchromatic Hubble Andromeda Treasury program. We use stellar population synthesis modeling to generate synthetic UV maps with projected spatial resolution of $sim$100 pc ($sim$24 arcseconds) The predicted UV flux agrees well with the observed flux, with median ratios between the modeled and observed flux of $log_{10}(f^{syn}/f^{obs}) = 0.03pm0.24$ and $-0.03pm0.16$ in the FUV and NUV, respectively. This agreement is particularly impressive given that we used only optical photometry to construct these UV maps. We use the dust-free maps to examine properties of obscured flux and star formation by comparing our reddened and dust-free FUV flux maps with the observed FUV and FUV+24{mu}m flux to examine the fraction of obscured flux. The synthetic flux maps require that $sim$90% of the FUV flux in M31 is obscured by dust, while the GALEX-based methods suggest that $sim$70% of the flux is obscured. This increase in the obscured flux estimate is driven by significant differences between the dust-free synthetic FUV flux and that derived when correcting the observed FUV for dust with 24{mu}m observations. The difference is further illustrated when we compare the SFRs derived from the FUV+24{mu}m flux with the 100 Myr average SFR from the SFHs. The 24{mu}m-corrected FUV flux underestimates the SFR by a factor of $sim$2.3 - 2.5. [abridged]
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