How to tell if a number is divisible by 7?

While studying for the GMAT, found this information and thought it would be good to share, seeing as how the test prep books all say that you can only find if a number is divisible by 7 through long division (Manhattan GMAT, Number Properties - updated for 12th Edition - page 14).



The source I'm quoting from was given to me by my wife.  She got the book in India:
The Pearson Guide to Objective Arithmetic (for competitive examinations) First Edition.  You can find the Third Edition here.

On pages 4 and 5, it breaks down the divisibility of numbers by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 25, 125, 18, and 88.

Here is what it says for Divisibility by 7:

The unit digit of the given number is doubled and then it is subtracted from the number obtained after omitting the unit digit.  If the remainder is divisible by 7, then the given number is also divisible by 7.
For example, consider 448.  On doubling the unit digit of 8, we get 16.
Then, 44 - 16 = 28.
Since 28 is divisible by 7, 448 is divisible by 7.

 The cool thing about this is that you can repeat the steps over and over again until you get your answer.  For example, take the number 3,752.

1. Separating the units digit, we get two numbers: 375 and 2.
1. 2 doubled is 4.
2. 375 - 4 = 371
2. Separating the units digit, we get two numbers: 37 and 1.
1. 1 doubled is 2.
2. 37 - 2 = 35.
3. We can stop here since we know 7 * 5 = 35, but if we want to, we can keep going.
3. Separating the units digit, we get  3 and 5.
1. 5 doubled is 10.
2. 3 - 10 = -7, which is divisible by 7.