An easy way to see if a number can be divided evenly by 11 is by following a few easy step. Choose a number. Add every other digit together. Then add the rest of the digits together. Subtract one of these numbers from the other. If the resulting number can be divided evenly by 11, then the original number can be divided by 11. Don't know if tye resulting number can divided evenly by 11? Repeat the step again and again until you get 0 or 11.
An example:
1212121416
+-+-+-+-+-+-
1+1+1+1+1=5
2+2+2+4+6=16
16-5=11. So 1212121416 can evenly divide by 11. When you divide by 11, the result is 110192856. Can we divide by 11 again?
1+0+9+8+6=24
1+1+2+5=9
24-5=19. So 110192856 does not divide evenly by 11. But there is more to this method than just decimal based numbers. This method works for all positive integer base 2 or greater, where we treat 11 as base+1!
11 in binary (base 2) is 3 in decimal (base 10). Applying this rule in base 2 will tell us if a number can be divided even by 3 in base 10. 10101 binary is 21 decimal.
1+1+1=11 in binary
0+0=0
11-0=11.
10101 divides evenly by 11 (binary)
21 divides evenly by 3 (decimal)
A14 base 12 is 1456 in decimal.
A+4=12
1=1
12-1=11. 1456 can divide by 13 evenly.
The 11 principle is one of my favorite discoveries, although I'm sure someone came across it before me. Thanks to whoever discovered this for decimal numbers, it helped me realize the rule works for other number bases too. :)