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Edit: I completely misread the post above. Anyway, here's some information on Hafiz which, while not relevant to this conversation, is pretty damn neat.
I think you would actually be suprised how many Muslims have memorized either large portions of the Qur'an, or the entire thing. There are actually schools for memorizing all ten methods of reciting the Qur'an. Unlike the Bible or the Torah, the Qur'an is written entirely in verse, which makes it much easier to memorize than most other holy books.
Here are four websites that teach people how to memorize the Qur'an and these were just the four top results from googling "Qur'an memorization."
http://www.houseofquran.com/quransys/qsys.html
http://www.houseofquran.com/ (mobile version. There are apps for everything from finding the direction of Mecca to memorization.)
https://www.tarteelequran.com/memorize-quran-online/
https://play.google.com/store/apps/details?id=com.engineery.memorizequran&hl=en
Here's more information of the practice of memorization. It has been very important in Islam since the religion's inception: