Cannot crop to an exact pixel dimensions

I am having a hard time cropping to an exact pixel dimensions upon export.  For my work, I need to crop to exactly 2166X1600, with a tolerance of no more than one pixel off of those dimensions, i.e. 2166X1599. 
I set crop tool to 2166X1600 and crop all of my images.  On the export I constrain to largest dimension of 2166 in both horizontal and vertical directions. Results vary widely from right on 2166X1600 to several pixels off on the short side.  Sometimes more than 1600, and sometimes less.  Any thoughts as to why these constraints do not work exactly?  Some sets of images I have to redo as many as 25% of a gallery of 200-300 images.  Any thoughts?  Thanks!

See, I don't need to crop the original image to 2166X1600.  I need a finished image at 2166X1600.  Meaning, in theory I should be able to crop an image to 1:1.35375 ratio, which is 2166X1600, and export to constrain to 2166 on the longest side.  When it's constrained to 2166, the short side should always be 1600.   
Here's where the problem is...  When you set the crop window to 2166X1600, you expect that when you size your crop on your image that the only options for the boundary are multiples of 2166X1600.  In reality, the crop visual boundaries are approximate.  Here's an example...
I have an image that I have cropped to a finished size of 3185X2350.  Which is based upon a 2166X1600 crop.  When I export it, I need the 3185 to become 2166.  So the long side is constrained to 2166, and it is reduced by a factor of 1.47.  3185/2166=1.47  If I reduce the short side, 2350, by that same factor of 1.47 you end up with a short side of 1598.  Which is why I have a problem.
My invalid assumption was that when you set specific pixel dimensions in the crop tool, that only those values would be shown in the crop tool's bounding box.  Photoshop can do it...

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