mosaic dem with multiple coordinate systems

I'm sure this is a basic question, but I'm struggling to mosaic multiple DEMs (~40) into one large raster. The DEMs are 30m NED rasters from the NRCS data gateway, for the New England region. They are in two different coordinate systems; NAD 1983 UTM Zone 18N and 19N.

I've read several posts debating whether or not DEMs should be projected or unprojected before running slope and hillshade tools. In the end, it sounds like running slope and hillshade on an unprojected mosaic of all DEMs would be most appropriate. That being said, I can't figure out how to create the mosaic dataset with DEMs from 2 different coordinate systems.

I have a VERY basic understanding of ArcGIS (using ArcGIS 10.2) and would appreciate any suggestions that could help mosaic the DEMs appropriately. Thanks.