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Division by 7, the Moura Velho Method

This week, Silvio Moura Velho showed me a new method for determining if a number is divisible
by 7 (and by 13). The method is described at:
http://www.7and13divisibility.com/.
I am impressed at how quickly the method works. What I feel is missing on
the site is a good explanation as to why it works. (The "Why it works" tab
seems to only provide another example.) Can you explain succinctly why
this method works?

Source: Original.

Solution

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