Shortcut: Divisible by 7 ?

Here's another one:

1. Lop off the last digit of the number in question. For 2016, that would be 6.

2. Double that number. Double the 6 to get 12.

3. Subtract that number from the remaining digits of the number in question, as if they were a stand-alone number. So you don't subtract 12 from 2016, or from 2010. You subtract 12 from 201 to get 189.

4. Repeat steps 1-3 until you get to a number you recognize as either a multiple of 7 or not. If you get a multiple of 7 after step 3, the original number was a multiple of 7. If you get a non-multiple of 7 after step 3, the original number was not a multiple of 7.

Take 189, lop of the 9, double to get 18, and subtract 18 from the 18 we got when we took 9 off the end of 189. Result 0 is divisible by 7, so 2016 also was.

Another example: 13,587

1. Separate: 1358 and 7
2. Double the 7: 14
3. Subtract: 1358-14 = 1344
1. Separate: 134 and 4
2. Double the 4: 8
3. Subtract: 134-8 = 126
Recognize 126 as 14 less than 140, i.e. a multiple of 7.

Conclusion: 13,587 is a multiple of 7.