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Per the Supported Raster Formats help, ArcMap does not write back out to .dem as the format is read only. You will definitely have to project the data between the coordinate systems, while you might be able to change the projection designation it will introduce new problems for the data. It will not display correctly since the extents were not altered, just what you defined it as.
There is an gdal utility that can process the conversion listed here. It looks as if you will need to convert the raster in ArcMap to an ASCII file prior to using it. I recommend the Raster to ASCII tool.
There are a lot of tools out there (3 modelling) that require USGS DEM input and getting your raster data into this format is not as straight forward as it might seem. As you probably already know this is something that ArcGIS does not do. I have personally not tried this but a lot of folks in the GIS SE community have mentioned the gdal_tranlate utility. Similar discussion here.
Original User: KevinCRDA
Thank you for your answers.
I have tried the GDAL libraries. It looks like there is a problem with the MTM projection. If I project my .DEM in UTM, and then after i use the gdal_translate librairy to go back to a USGS DEM file, it works well. But if I project my original DEM in the MTM projection, when I use the gdal_translate library it always give me the same error:
Input file size is 1201, 1201
ERROR 1: Too many points (441 out of 441) failed to transform,
unable to compute output bounds.
Any ideas on why it doesn't work with MTM ?