CUDA Toolkit very finicky about Visual Studio

An unexpected gotcha with nVidia CUDA Toolkit on Windows is that it refuses to work with the current version of Microsoft Visual Studio:

nvcc fatal   : nvcc cannot find a supported version of Microsoft Visual Studio. Only the versions 2010, 2012, and 2013 are supported

According to the installation guide (and the above compile error), CUDA Toolkit 7.5 only integrates with Visual Studio 2013, 2012 or 2010. I could have saved myself many hours of downloading and installing Visual Studio 2015 and Intel Parallel Studio XE 16.0, if I’d known that beforehand! Now to uninstall everything and start again from scratch…