Clever but most people won't remember these tricks. I for one would appreciate a refresher on the other divisibility rules/tricks.
Divisibility by 2: Last digit is divisible by 2 (0, 2, 4, 6, or 8).
Divisibility by 3: Sum of digits is divisible by 3. (You can repeatedly sum the digits until the sum is small enough to manage easily.)
Divisibility by 4: Last two digits are divisible by 4 (00, 04, 08, 12, ...). If you don't know your multiples of four, you can divide the two digits by 2 and then see if it's even.
Divisibility by 5: Last digit is divisible by 5 (0 or 5).
Divisibility by 6: Divisible by 2 and by 3. (Works similarly for all composite numbers, of course. Like 10, 14, 15, etc. Make sure the factors you're checking are relatively prime (share no common factors), or it won't work. For example, divisibility by 2 and 6 does not imply divisibility by 12.)
Divisibility by 8: Last three digits are divisible by 8 (000, 008, 016, ...). Probably faster to divide the last three by 2 at least once then check divisibility by 4.
Divisibility by 9: Sum of digits is divisible by 9. (You can repeatedly sum the digits until the sum is small enough to manage easily.)
Divisibility by 11: Alternating sum of digits is divisible by 11. (Almost always turns out to be -11, 0, or 11. Example: 66374. 6 - 6 + 3 - 7 + 4 = 0, so 66374 is divisible by 11.)
Division is the most 'complicated' of the arithmetic operations and if you rely on doing sums in your head frequently then knowing such shortcuts can speed up the calculations considerably.
It also helps in checking results of other calculations.
Do primes come in degrees these days?
* 2 - if it ends in 0,2,4,6,8, it's divisible by 2
* 3 - if the sum of the digits is divisible by 3, so is the number
* 4 - Ignore everything but the last 2 digits. If the last 2 digits make a number that's divisible by 4, the whole number is (eg. anything ending in 52 is divisible by 4 because 52 is divisible by 4)
* 5 - If it ends in 0 or 5, it's divisible by 5.
* 6 - do an and of the 2 and 3 tests.
* 8 - Same as the 4 test, only last 3 digits.
* 9 - same as 3, only the digit sum needs to be divisible by 9 instead of 3
* 10 - If it ends in 0, it's divisible by 10.
So I guess the only one I don't know is 11.
5060 is an exception to your rule.
As described before, if (even digit sum) - (odd digit sum) is divisible by 11, then the number is divisible by 11.