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The MacTeX-2016 package can be obtained over the internet for free by clicking on the following link:

MacTeX.pkg [ approximately 2.5G - 14 June 2015 ]
This link points to a "multiplexer" site which parcels out requests to various up-to-date CTAN mirrors. Occasionally users have trouble getting the package from this multiplexer. In that case, choose a specific mirror on this mirror page.

Was the download successful?

There is an easy way to check that your download was successful. Run Apple's Terminal application in /Applications/Utilities. After the prompt, type the command
openssl md5  
and add a space at the end. Then drag and drop the file MacTeX.pkg to the Terminal window to complete the command. Push Return.
The Mac will compute an "md5 checksum". This sum should be: bf579a512d31253d828591fdb3644dca

What to do if the Checksum is bad

Many users report trouble after downloading with Google Chrome, and possibly other browsers. We strongly recommend downloading with Safari. Users with problems also report success after downloading via the torrent network.

What to do if the Installation Succeeds but TeX Doesn't Work

In this case, read the questions and answers in the Frequently Asked Questions page on "getting or installing MacTeX"

TUG home page; (via Google)